دانلود کتاب Measure Theory 1-5

فهرست مطالب :

Measure Theory 1_The Irreducible Minimum(2e,2011,102p)D.H.Fremlin_9780953812981
Contents
General Introduction
Introduction to Volume 1
Note on second and third printings
Note on second edition, 2011
Chapter 11. Measure spaces
111. σ-algebras
111A. Definition
111B. Remarks
111C. Infinite unions and intersections
111D. Elementary properties of σ-algebras
111E. More on infinite unions and intersections
111F. Countable sets
111G. Borel sets
111X. Basic exercises
111Y. Further exercises
111 Notes and comments
112. Measure spaces
112A. Definition
112B. Remarks
112C. Elementary properties of measure spaces
112D. Negligible sets
112X. Basic exercises
112Y. Further exercises
112 Notes and comments
113. Outer measures and Carath éodory\'s construction
113A. Outer measures
113B. Remarks
113C. Carathéodory\'s Method
113D. Remark
113X. Basic exercises
113Y. Further exercises
113 Notes and comments
114. Lebesgue measure on R
114A. Definitions
114B. Lemma
114C. Definition
114D. Proposition
114E. Definition
114F. Lemma
114G. Proposition
114X. Basic exercises
114Y. Further exercises
114 Notes and comments
115. Lebesgue measure on R^r
115A. Definitions
115B. Lemma
115C. Definition
115D. Proposition
115E. Definition
115F. Lemma
115G. Proposition
115X. Basic exercises
115Y. Further exercises
115 Notes and comments
Chapter 12. Integration
121. Measurable functions
121A. Lemma
121B. Proposition
121C. Definition
121D. Proposition
121E. Theorem
121F. Theorem
121G. Remarks
121H. Proposition
*121I.
*121J.
*121K. Proposition
121X. Basic exercises
121Y. Further exercises
121 Notes and comments
122. Definition of the integral
122A. Definitions
122B. Lemma
122C. Lemma
122D. Corollary
122E. Definition
122F. Proposition
122G. Lemma
122H. Definition
122I. Lemma
122J. Lemma
122K. Definition
122L. Lemma
122M. Definition
122N. Remarks
122O. Theorem
122P. Theorem
122Q. Remark
122R. Corollary
122X. Basic exercises
122Y. Further exercises
122 Notes and comments
123. The convergence theorems
123A. B.Levi\'s theorem
123B. Fatou\'s Lemma
123C. Lebesgue\'s Dominated Convergence Theorem
123D. differentiating through an integral
123X. Basic exercises
123Y. Further exercises
123 Notes and comments
Chapter 13. Complements
131. Measurable subspaces
131A. Proposition
131B. Definition
131C. Lemma
131D. Integration over subsets: Definition
131E. Proposition
131F. Corollary
131G. Corollary
131H. Corollary
131X. Basic exercises
131Y. Further exercises
131 Notes and comments
132. Outer measures from measures
132A. Proposition
132B. Definition
132C. Proposition
132D. Measurable envelopes
132E. Lemma
132F. Full outer measure
132X. Basic exercises
132Y. Further exercises
132 Notes and comments
133. Wider concepts of integration
133A. Infinite integrals
133B. Functions with exceptional values
133C. Complex-valued functions
133D. Definitions
133E. Lemma
133F. Proposition
133G. Lebesgue\'s Dominated Convergence Theorem
133H. Corollary
133I. Upper and lower integrals
133J. Proposition
133K. Convergence theorems for upper integrals
*133L.
133X. Basic exercises
133Y. Further exercises
133 Notes and comments
134. More on Lebesgue measure
134A. Proposition
134B. Theorem
*134C. Remark
*134D.
134E. Borel sets and Lebesgue measure on R^r
134F. Proposition
134G. The Cantor set
134H. The Cantor function
134I. The Cantor function modified
134J. More examples
*134K. Riemann integration
*134L.
134X. Basic exercises
134Y. Further exercises
134 Notes and comments
135. The extended real line
135A. The algebraic structure of [-∞, ∞]
135B. The order structure of [-∞,. ∞]
135C. The Borel structure of [-∞,. ∞]
135D. Convergent sequences in [-∞,. ∞]
135E. Measurable functions
135F. [-∞,. ∞]-valued integrable functions
135G.
135H. Upper and lower integrals again
135I. Subspace measures
135X. Basic exercises
135Y. Further exercises
135 Notes and comments
*136. The Monotone Class Theorem
136A. Lemma
136B. Monotone Class Theorem
136C. Corollary
136D. Corollary
136E. Algebras of sets: Definition
136F. Remarks
136G. Theorem
*136H. Proposition
136X. Basic exercises
136Y. Further exercises
136 Notes and comments
Appendix to Volume 1 - Useful Facts
1A1. Set theory
1A1A. Square bracket notations
1A1B. Direct and inverse images
1A1C. Countable sets
1A1D. Proposition
1A1E. Properties of countable sets
1A1F.
*1A1G. Remark
1A1H. Some uncountable sets
1A1I. Remark
1A1J. Notation
1A1 Notes and comments
1A2. Open and closed sets in R^r
1A2A. Open sets
1A2B. The family of all open sets
1A2C. Cauchy\'s inequality
1A2D. Corollary
1A2E Closed sets
1A2F. Proposition
1A2G.
1A3. Lim sups and lim infs
1A3A. Definition
1A3B.
1A3C. Remark
*1A3D. Other expressions of the same idea
Concordance
References for Volume 1
Index to volume 1
Principal topics and results
General index
Measure Theory 2_Broad Foundations(2016,570p)D.H.Fremlin_9780953812974
Contents
General introduction
Introduction to Volume 2
Note on second printing, April 2003
Note on hardback edition, January 2010
Note on second printing of hardback edition, April 2016
*Chapter 21. Taxonomy of measure spaces
211. Definitions
211A. Definition
211B. Definition
211C. Definition
211D. Definition
211E. Definition
211F. Definition
211G. Definition
211H. Definition
211I. Definition
211J. Definition
211K. Definition
211L.
211M. Example: Lebesgue measure
211N. Counting measure
211O. A non-semi-finite space
211P. A non-complete space
211Q. Some probability spaces
211R. Countable-cocountable measure
211X. Basic exercises
211Y. Further exercises
211 Notes and comments
212. Complete spaces
212A. Proposition
212B. Proposition
212C. The completion of a measure
212D.
212E.
212F.
212G.
212X. Basic exercises
212Y. Further exercises
212 Notes and comments
213. Semi-finite, locally determined and localizable spaces
213A. Lemma
213B. Proposition
*213C. Proposition
213D. C.l.d. versions
213E. Definition
213F.
213G.
213H.
213I. Locally determined negligible sets
213J. Proposition
*213K. Lemma
213L. Proposition
213M. Corollary
213N.
213O.
213X. Basic exercises
213Y. Further exercises
213 Notes and comments
214. Subspaces
214A. Proposition
214B. Definition
214C. Lemma
214D. Integration over subsets
214E. Proposition
214F. Proposition
214G. Corollary
214H. Subspaces and Carathéodory\'s method
214I.
214J. Upper and lower integrals
214K. Measurable subspaces
214L. Direct sums
214M. Proposition
214N. Corollary
*214O.
*214P. Theorem
*214Q. Proposition
214X. Basic exercises
214Y. Further exercises
214 Notes and comments
215 σ-finite spaces and the principle of exhaustion
215A. The principle of exhaustion
215B.
215C. Corollary
215D.
*215E.
215X. Basic exercises
215Y. Further exercises
215 Notes and comments
216. Examples
216A. Lebesgue measure
216B.
*216C. A complete, localizable, non-locally-determined space
*216D. A complete, locally determined space which is not localizable
*216E. A complete, locally determined, localizable space which is not strictly localizable
216X. Basic exercises
216Y. Further exercises
216 Notes and comments
Chapter 22. The Fundamental Theorem of Calculus
221. Vitali\'s theorem in R
221A. Vitali\'s theorem
221B. Remarks
221X. Basic exercises
221Y. Further exercises
221 Notes and comments
222. Differentiating an indefinite integral
222A. Theorem
222B. Remarks
222C. Lemma
222D. Lemma
222E. Theorem
222F. Corollary
222G. Corollary
222H.
222I. Complex-valued functions
*222J. The Denjoy-Young-Saks theorem
*222K. Lemma
*222L. Theorem
222X. Basic exercises
222Y. Further exercises
222 Notes and comments
223. Lebesgue\'s density theorems
223A. Lebesgue\'s Density Theorem
223B. Corollary
223C. Corollary
223D. Theorem
223E. Complex-valued functions
223X. Basic exercises
223Y. Further exercises
223 Notes and comments
224. Functions of bounded variation
224A. Definition
224B. Remarks
224C. Proposition
224D. Theorem
224E. Corollary
224F. Corollary
224G. Corollary
224H. Proposition
224I. Theorem
224J.
224K. Complex-valued functions
224X. Basic exercises
224Y. Further exercises
224 Notes and comments
225. Absolutely continuous functions
225A. Absolute continuity of the indefinite integral
225B. Absolutely continuous functions on R
225C. Proposition
225D. Lemma
225E. Theorem
225F. Integration by parts
225G.
225H. Semi-continuous functions
225I. Proposition
225J.
225K. Proposition
225L. Corollary
225M. Corollary
225N. The Cantor function
225O. Complex-valued functions
225X. Basic exercises
225Y. Further exercises
225 Notes and comments
226. The Lebesgue decomposition of a function of bounded variation
226A. Sums over arbitrary index sets
226B. Saltus functions
226C. The Lebesgue decomposition of a function of bounded variation
226D. Complex functions
226E.
226X. Basic exercises
226Y. Further exercises
226 Notes and comments
Chapter 23. The Radon-Nikodym Theorem
231. Countably additive functionals
231A. Definition
231B. Elementary facts
231C. Definition
231D. Elementary facts
231E.
231F. Corollary
231X. Basic exercises
231Y. Further exercises
231 Notes and comments
232. The Radon-Nikodým theorem
232A. Absolutely continuous functionals
232B. Proposition
232C. Lemma
232D. Proposition
232E. The Radon-Nikodým theorem
232F. Corollary
232G. Corollary
232H. Remarks
232I. The Lebesgue decomposition of a countably additive functional
232X. Basic exercises
232Y. Further exercises
232 Notes and comments
233. Conditional expectations
233A. σ-subalgebras
233B. Lemma
233C. Remarks
233D. Conditional expectations
233E.
233F. Remarks
233G. Convex functions
233H.
233I. Jensen\'s inequality
233J.
233K.
233X. Basic exercises
233Y. Further exercises
233 Notes and comments
234. Operations on measures
234A. Inverse-measure-preserving functions
234B. Proposition
234C. Image measures
234D. Definition
234E. Proposition
*234F.
234G. Sums of measures
234H. Proposition
234I. Indefinite-integral measures
234J. Definition
234K. Remarks
234L. The domain of an indefinite-integral measure
234M. Corollary
*234N.
*234O.
234P. Ordering measures
234Q. Proposition
234X. Basic exercises
234Y. Further exercises
234 Notes and comments
235. Measurable transformations
235A.
235B. Remarks
235C.
235D.
235E.
235F. Remarks
235G. Theorem
235H. The image measure catastrophe
235I. Lemma
235J. Theorem
235K. Corollary
235L. Applying the Radon-Nikodým theorem
235M. Theorem
235N. Remark
*235O.
*235P. Proposition
*235Q.
235R. Reversing the burden
235X. Basic exercises
235Y. Further exercises
235 Notes and comments
Chapter 24. Function spaces
241. \\usepackage{euscript} \\mathscr{L}^0 and L^0
241A. The space \\usepackage{euscript} \\mathscr{L}^0 : Definition
241B. Basic properties
241C. The space L^0: Definition
241D. The linear structure of L^0
241E. The order structure of L^0
241F. Riesz spaces
241G.
241H. The multiplicative structure of L^0
241I. The action of Borel functions on L^0
241J. Complex L^0
241X. Basic exercises
241Y. Further exercises
241 Notes and comments
242. L^1
242A. The space L^1
242B. Theorem
242C. The order structure of L^1
242D. The norm of L^1
242E.
242F. Theorem
242G. Definition
242H. L^1 as a Riesz space
242I. The Radon-Nikodým theorem
242J. Conditional expectations revisited
242K.
242L. Proposition
242M. L^1 as a completion
242N.
242O. Theorem
242P. Complex L^1
242X. Basic exercises
242Y. Further exercises
242 Notes and comments
243. L^∞
243A. Definitions
243B. Theorem
243C. The order structure of L^∞
243D The norm of L^∞
243E. Theorem
243F. The duality between L^∞ and L^1
243G. Theorem
243H.
243I. A dense subspace of L^∞
243J. Conditional expectations
243K. Complex L^∞
243X. Basic exercises
243Y. Further exercises
243 Notes and comments
244. L^p
244A. Definitions
244B. Theorem
244C. The order structure of L^p
244D. The norm of L^p
244E.
244F. Proposition
244G. Theorem
244H.
*244I. Corollary
244J. Duality in L^p
244K. Theorem
244L.
244M.
244N. The space L^2
*244O.
244P. Complex L^p
244X. Basic exercises
244Y. Further exercises
244 Notes and comments
245. Convergence in measure
245A. Definitions
245B. Remarks
245C. Pointwise convergence
245D. Proposition
245E.
245F. Alternative description of the topology of convergence in measure
245G. Embedding L^p in L^0
245H.
245I. Remarks
245J.
245K.
245L. Corollary
245M. Complex L^0
245X. Basic exercises
245Y. Further exercises
245 Notes and comments
246. Uniform integrability
246A. Definition
246B. Remarks
246C.
246D. Proposition
246E. Remarks
246F.
246G.
246H. Remarks
246I. Corollary
246J.
246K. Complex \\usepackage{euscript} \\mathscr{L}^1 and L^1
246X. Basic exercises
246Y. Further exercises
246 Notes and comments
247. Weak compactness in L^1
247A.
247B. Corollary
247C. Theorem
247D. Corollary
247E. Complex L^1
247X. Basic exercises
247Y. Further exercises
247 Notes and comments
Chapter 25. Product Measures
251. Finite products
251A. Definition
251B. Lemma
251C. Definition
251D. Definition
251E. Proposition
251F. Definition
251G. Remark
251H.
251I.
251J. Proposition
251K. σ-finite spaces
*251L.
251M.
251N. Theorem
251O.
251P. Lemma
251Q. Proposition
251R. Corollary
251S. Corollary
251T.
251U.
*251W. Products of more than two spaces
251X. Basic exercises
251Y. Further exercises
251 Notes and comments
252. Fubini\'s theorem
252A. Repeated integrals
252B. Theorem
252C.
252D.
252E. Corollary
252F.
252G.
252H. Corollary
252I. Corollary
252J. Remarks
252K. Example
252L. Example
252M. Remark
252N. Integration through ordinate sets I
252O. Integration through ordinate sets II
*252P.
252Q. The volume of a ball
252R. Complex-valued functions
252X. Basic exercises
252Y. Further exercises
252 Notes and comments
253. Tensor products
253A. Bilinear operators
253B. Definition
253C. Proposition
253D.
253E. The canonical map L^1 × L^1 -> L^1
253F.
253G. The order structure of L^1
253H. Conditional expectations
253I.
*253J. Upper integrals
*253K.
253L. Complex spaces
253X. Basic exercises
253Y. Further exercises
253 Notes and comments
254. Infinite products
254A. Definitions
254B. Lemma
254C. Definition
254D. Remarks
254E. Definition
254F. Theorem
254G.
254H. Corollary
254I.
254J. The product measure on {0,1}^I
254K.
254L. Subspaces
254M.
254N. Theorem
254O. Proposition
254P. Proposition
254Q. Proposition
254R. Conditional expectations again
254S. Proposition
254T. Remarks
*254U.
*254V.
254X. Basic exercises
254Y. Further exercises
254 Notes and comments
255. Convolutions of functions
255A.
255B. Corollary
255C. Remarks
255D.
255E. The basic formula
255F. Elementary properties
255G.
255H.
255I. Corollary
255J. Theorem
255K.
255L. The r-dimensional case
255M. The case of ]-π, π]
255N. Theorem
255O. Convolutions on ]-π, π]
255X. Basic exercises
255Y. Further exercises
255 Notes and comments
256. Radon measures on R^r
256A. Definitions
256B.
256C. Theorem
256D. Proposition
256E.
256F. Theorem
256G. Theorem
256H. Examples
256I. Remarks
256J. Absolutely continuous Radon measures
256K. Products
256L. Remark
*256M.
256X. Basic exercises
256Y. Further exercises
256 Notes and comments
257. Convolutions of measures
257A. Definition
257B. Theorem
257C. Corollary
257D. Corollary
257E. Corollary
257F. Theorem
257X. Basic exercises
257Y. Further exercises
257 Notes and comments
Chapter 26. Change of Variable in the Integral
261. Vitali\'s theorem in R^r
261A. Notation
261B. Vitali\'s theorem in R^r
261C.
261D. Corollary
261E. Theorem
261F.
261X. Basic exercises
261Y. Further exercises
261 Notes and comments
262 Lipschitz and differentiable functions
262A. Lipschitz functions
262B.
262C. Remark
262D. Proposition
262E. Corollary
262F. Differentiability
262G. Remarks
262H. The norm of a matrix
262I. Lemma
262J. Remark
262K. The Cantor function revisited
262L.
262M.
262N. Corollary
262O. Corollary
262P. Corollary
*262Q.
262X. Basic exercises
262Y. Further exercises
262 Notes and comments
263. Differentiable transformations in R^r
263A. Linear transformations
263B. Remark
263C. Lemma
263D.
263E. Remarks
*263F. Corollary
263G. Polar coordinates in the plane
263H. Corollary
263I.
263J. The one-dimensional case
263X. Basic exercises
263Y. Further exercises
263 Notes and comments
264 Hausdorff measures
264A. Definitions
264B.
264C. Definition
264D. Remarks
264E. Theorem
264F. Proposition
264G. Lipschitz functions
264H.
264I. Theorem
*264J. The Cantor set
264X. Basic exercises
264Y. Further exercises
264 Notes and comments
265. Surface measures
265A. Normalized Hausdorff measure
265B. Linear subspaces
265C. Corollary
265D.
265E. Theorem
265F. The surface of a sphere
265G.
265H. Corollary
265X. Basic exercises
265Y. Further exercises
265 Notes and comments
*266 .The Brunn-Minkowski inequality
266A. Arithmetic and geometric means
266B. Proposition
266C. Theorem
266X. Basic exercises
266 Notes and comments
Chapter 27. Probability theory
271. Distributions
271A. Notation
271B. Theorem
271C. Definition
271D. Remarks
271E. Measurable functions of random variables
271F. Corollary
271G. Distribution functions
271H. Densities
271I. Proposition
271J.
271K.
*271L.
271X. Basic exercises
271Y. Further exercises
271 Notes and comments
272. Independence
272A. Definitions
272B. Remarks
272C. The σ-subalgebra defined by a random variable
272D. Proposition
272E. Corollary
272F.
272G. Distributions of independent random variables
272H. Corollary
272I. Corollary
272J.
272K. Proposition
272L.
272M. Products of probability spaces and independent families of random variables
272N.
272O. Tail σ-algebras and the zero-one law
272P.
*272Q.
272R.
272S. Bienaymé\'s Equality
272T. The distribution of a sum of independent random variables
272U. Corollary
272V.
*272W.
272X. Basic exercises
272Y. Further exercises
272 Notes and comments
273. The strong law of large numbers
273A.
273B. Lemma
273C.
273D. The strong law of large numbers: first form
273E. Corollary
273F. Corollary
273G. Corollary
273H. Strong law of large numbers: second form
273I. Strong law of large numbers: third form
273J. Corollary
273K.
273L.
*273M.
273N. Theorem
273X. Basic exercises
273Y. Further exercises
273 Notes and comments
274. The central limit theorem
274A. The normal distribution
274B. Proposition
274C. Lemma
274D. Lemma
274E. Lemma
274F. Lindeberg\'s theorem
274G. Central Limit Theorem
274H. Remarks
274I. Corollary
274J. Corollary
274K. Corollary
274L. Remarks
*274M.
274X. Basic exercises
274Y. Further exercises
274 Notes and comments
275. Martingales
275A. Definition
275B. Examples
275C. Remarks
275D.
275E. Up-crossings
275F. Lemma
275G.
275H. Theorem
275I. Theorem
*275J.
275K. Reverse martingales
275L. Stopping times
275M. Examples
275N. Lemma
275O. Proposition
275P. Corollary
275X. Basic exercises
275Y. Further exercises
275 Notes and comments
276. Martingale difference sequences
276A. Martingale difference sequences
276B. Proposition
276C. The strong law of large numbers: fourth form
276D. Corollary
276E. `Impossibility of systems\'
*276F.
*276G. Lemma
*276H. Komlós\' theorem (KOMLÓS 67)
276X. Basic exercises
276Y. Further exercises
276 Notes and comments
Chapter 28. Fourier analysis
281. The Stone-Weierstrass theorem
281A. Stone-Weierstrass theorem: first form
281B.
281C. Lemma
281D. Corollary
281E. Stone-Weierstrass theorem: second form
281F. Corollary: Weierstrass\' theorem
281G. Stone-Weierstrass theorem: third form
281H. Corollary
281I. Corollary
281J. Corollary
281K. Corollary
281L. Corollary
281M. Weyl\'s Equidistribution Theorem
281N. Theorem
281X. Basic exercises
281Y. Further exercises
281 Notes and comments
282. Fourier series
282A. Definition
282B. Remarks
282C. The problems
282D. Lemma
282E.
282F. Corollary
282G.
282H.
282I. Corollary
282J.
282K. Corollary
282L.
282M. Lemma
282N.
282O. Theorem
282P. Corollary
282Q.
*282R.
282X. Basic exercises
282Y. Further exercises
282 Notes and comments
283. Fourier transforms I
283A. Definitions
283B. Remarks
283C. Proposition
283D. Lemma
283E.
283F. Theorem
283G. Corollary
283H. Lemma
283I. Theorem
283J. Corollary
283K.
283L.
283M.
283N.
283O.
283W. Higher dimensions
283X. Basic exercises
283Y. Further exercises
283 Notes and comments
284. Fourier transforms II
284A. Test functions: Definition
284B.
284C. Proposition
284D. Definition
284E.
284F.
284G. Lemma
284H. Definition
284I. Remarks
284J. Lemma
284K. Proposition
284L.
284M. Theorem
284N. L^2 spaces
284O. Theorem
284P. Corollary
284Q. Remarks
284R. Dirac\'s delta function
284W. The multidimensional case
284X. Basic exercises
284Y. Further exercises
284 Notes and comments
285. Characteristic functions
285A. Definition
285B. Remarks
285C.
285D.
285E. Lemma
285F.
285G. Corollary
285H. Remark
285I. Proposition
285J.
285K. Characteristic functions and the vague topology
285L. Theorem
285M. Corollary
285N. Remarks
285O. Lemma
285P. Lemma
285Q. Law of Rare Events: Theorem
285R. Convolutions
285S. The vague topology and pointwise convergence of characteristic functions
285T. Proposition
285U. Corollary
285X. Basic exercises
285Y. Further exercises
285 Notes and comments
286. Carleson\'s theorem
286A. The Maximal Theorem
286B. Lemma
286C. Shift, modulation and dilation
286D. Lemma
286E. The Lacey-Thiele construction
286F. A partial order
286G.
286H. `Mass\' and `energy\' (Lacey & Thiele 00)
286I. Lemma
286J. Lemma
286K. Lemma
286L. Lemma
286M. The Lacey-Thiele lemma
286N. Lemma
286O. Lemma
286P. Lemma
286Q. Lemma
286R. Lemma
286S. Lemma
286T. Lemma
286U. Theorem
286V. Theorem
286W. Glossary
286X. Basic exercises
286Y. Further exercises
286 Notes and comments
Appendix to Volume 2 - Useful Facts
2A1. Set theory
2A1A. Ordered sets
2A1B. Transfinite Recursion: Theorem
2A1C. Ordinals
2A1D. Basic facts about ordinals
2A1E. Initial ordinals An initial ordinal
2A1F. Basic facts about initial ordinals
2A1G. Schröder-Bernstein theorem
2A1H. Countable subsets of PN
2A1I. Filters
2A1J. The Axiom of Choice
2A1K. Zermelo\'s Well-Ordering Theorem
2A1L. Fundamental consequences of the Axiom of Choice
2A1M. Zorn\'s Lemma
2A1N. Ultrafilters
2A1O. The Ultrafilter Theorem
2A1P.
2A2. The topology of Euclidean space
2A2A. Closures: Definition
2A2B. Lemma
2A2C. Continuous functions
2A2D. Compactness in R^r: Definition
2A2E. Elementary properties of compact sets
2A2F.
2A2G. Corollary
2A2H. Lim sup and lim inf revisited
2A2I.
2A3. General topology
2A3A. Topologies
2A3B. Continuous functions
2A3C. Subspace topologies
2A3D. Closures and interiors
2A3E Hausdorff topologies
2A3F. Pseudometrics
2A3G. Proposition
2A3H.
2A3I. Remarks
2A3J. Subspaces: Proposition
2A3K. Closures and interiors
2A3L. Hausdorff topologies
2A3M. Convergence of sequences
2A3N. Compactness
2A3O. Cluster points
2A3P. Filters
2A3Q. Convergent filters
2A3R.
2A3S. Further calculations with filters
2A3T. Product topologies
2A3U. Dense sets
2A4. Normed spaces
2A4A. The real and complex fields
2A4B. Definitions
2A4C. Linear subspaces
2A4D. Banach spaces
2A4E.
2A4F. Bounded linear operators
2A4G. Theorem
2A4H. Dual spaces
2A4I. Extensions of bounded operators: Theorem
2A4J. Normed algebras
*2A4K. Definition
2A5. Linear topological spaces
2A5A. Linear space topologies
2A5B.
*2A5C.
2A5D. Definition
2A5E. Convex sets
2A5F. Completeness in linear topological spaces
2A5G.
2A5H. Normed spaces and sequential completeness
2A5I. Weak topologies
*2A5J. Angelic spaces
2A6. Factorization of matrices
2A6A. Determinants
2A6B. Orthonormal families
2A6C.
Concordance
References for Volume 2
Index to volumes 1 and 2
Principal topics and results
General index
Measure Theory 3-1_Measure Algebras(2e,2012,216p)D.H.Fremlin_9780956607102
Contents
General introduction
Introduction to Volume 3
Note on second printing
Note on second (`Lulu\') edition
Chapter 31. Boolean algebras
311. Boolean algebras
311A. Definitions
311B. Examples
311C. Proposition
311D. Lemma
311E. M.H.Stone\'s theorem: first form
311F. Remarks
311G. The operations ∪, ＼, Δ on a Boolean ring
311H. The order structure of a Boolean ring
311I. The topology of a Stone space: Theorem
311J.
311K. Remark
311L. Complemented distributive lattices
311X. Basic exercises
311Y. Further exercises
311 Notes and comments
312. Homomorphisms
312A. Subalgebras
312B. Proposition
312C. Ideals in Boolean algebras: Proposition
312D. Principal ideals
312E. Proposition
312F. Boolean homomorphisms
312G. Proposition
312H. Proposition
312I. Proposition
312J. Proposition
*312K. Fixed-point subalgebras
312L. Quotient algebras: Proposition
312M.
312N.
312O. Lemma
312P. Homomorphisms and Stone spaces
312Q. Theorem
312R. Theorem
312S. Proposition
312T. Principal ideals
312X. Basic exercises
312Y. Further exercises
312 Notes and comments
313. Order-continuous homomorphisms
313A. Relative complementation: Proposition
313B. General distributive laws: Proposition
313C.
313D.
313E. Order-closed subalgebras and ideals
313F. Order-closures and generated sets
313G.
313H. Definitions
313I. Proposition
313J.
313K. Lemma
313L. Proposition
313M.
313N. Definition
313O. Proposition
313P.
313Q. Corollary
313R.
313S. Upper envelopes
313X. Basic exercises
313Y. Further exercises
313 Notes and comments
314. Order-completeness
314A. Definitions
314B. Remarks
314C. Proposition
314D. Corollary
314E. Proposition
314F.
314G. Corollary
314H. Corollary
314I. Corollary
314J.
314K. Extension of homomorphisms
314L. The Loomis-Sikorski representation of a Dedekind σ-complete Boolean algebra
314M. Theorem
314N. Corollary
314O. Regular open algebras
314P. Theorem
314Q. Remarks
*314R.
314S.
314T.
314U. The Dedekind completion of a Boolean algebra
314X. Basic exercises
314Y. Further exercises
314 Notes and comments
315. Products and free products
315A. Products of Boolean algebras
315B. Theorem
315C. Products of partially ordered sets
315D. Proposition
315E. Factor algebras as principal ideals
315F. Proposition
315G. Algebras of sets and their quotients
*315H.
315I. Free products
315J. Theorem
315K.
315L. Proposition
315M. Algebras of sets and their quotients
315N. Notation
315O. Lemma
315P. Example
315Q. Example
*315R. Projective and inductive limits: Proposition
*315S. Definitions
315X. Basic exercises
315Y. Further exercises
315 Notes and comments
316. Further topics
316A. Definitions
316B. Theorem
316C. Proposition
316D. Corollary
316E. Proposition
316F. Corollary
316G. Definition
316H. Proposition
316I.
316J. The regular open algebra of R
316K. Atoms in Boolean algebras
316L. Proposition
316M. Proposition
316N. Definition
*316O. Lemma
*316P. Proposition
*316Q. Proposition
316X. Basic exercises
316Y. Further exercises
316 Notes and comments
Chapter 32. Measure algebras
321. Measure algebras
321A. Definition
321B. Elementary properties of measure algebras
321C. Proposition
321D. Corollary
321E. Corollary
321F. Corollary
321G. Subalgebras
321H. The measure algebra of a measure space
321I. Definition
321J. The Stone representation of a measure algebra
321K. Definition
321X. Basic exercises
321Y. Further exercises
321 Notes and comments
322. Taxonomy of measure algebras
322A. Definitions
322B.
322C.
322D.
322E. Proposition
322F. Proposition
322G.
322H. Principal ideals
322I. Subspace measures
322J. Corollary
322K. Indefinite-integral measures
322L. Simple products
*322M. Strictly localizable spaces
322N. Subalgebras
322O. The Stone space of a localizable measure algebra
322P. Theorem
322Q. Definition
322R. Further properties of Stone spaces
322X. Basic exercises
322Y. Further exercises
322 Notes and comments
323. The topology of a measure algebra
323A. The pseudometrics
323B. Proposition
323C. Proposition
323D.
323E. Corollary
323F.
323G. The classification of measure algebras
323H. Closed subalgebras
323I. Notation
323J. Proposition
323K.
323L. Proposition
*323M.
323X. Basic exercises
323Y. Further exercises
323 Notes and comments
324. Homomorphisms
324A. Theorem
324B. Corollary
324C. Remarks
324D. Proposition
324E. Stone spaces
324F.
324G. Corollary
324H. Corollary
324I. Definition
324J. Proposition
324K. Proposition
324L. Corollary
324M. Proposition
324N. Proposition
324O. Proposition
*324P.
324X. Basic exercises
324Y. Further exercises
324 Notes and comments
325. Free products and product measures
325A. Theorem
325B. Characterizing the measure algebra of a product space
325C.
325D. Theorem
325E. Remarks
325F.
325G.
*325H. Products of more than two factors
325I. Infinite products
325J.
325K. Definition
325L. Independent subalgebras
325M.
*325N. Notation
325X. Basic exercises
325Y. Further exercises
325 Notes and comments
326. Additive functionals on Boolean algebras
326A. Additive functionals
326B. Elementary facts
326C. The space of additive functionals
326D. The Jordan decomposition (I)
*326E. Additive functionals on free products
*326F.
*326G. Lemma
*326H Liapounoff\'s convexity theorem (LIAPOUNOFF 40)
326I. Countably additive functionals
326J. Elementary facts
326K. Corollary
326L. The Jordan decomposition (II)
326M. The Hahn decomposition
326N. Completely additive functionals
326O. Basic facts
326P.
326Q. The Jordan decomposition (III)
326R.
326S.
326T. Corollary
326X. Basic exercises
326Y. Further exercises
326 Notes and comments
327. Additive functionals on measure algebras
327A.
327B. Theorem
327C. Proposition
327D. The Radon-Nikodým theorem
327E.
327F. Standard extensions
327G. Definition
327X. Basic exercises
327Y. Further exercises
327 Notes and comments
*328 .Reduced products and other constructions
328A. Construction
328B. Proposition
328C. Definition
328D. Proposition
328E. Proposition
328F. Corollary
328G. Corollary
328H. Proposition
328I.
328J.
328X. Basic exercises
328 Notes and comments
Chapter 33. Maharam\'s theorem
331. Maharam types and homogeneous measure algebras
331A. Definition
331B.
331C. Corollary
331D. Lemma
331E. Generating sets
331F. Maharam types
331G.
331H. Proposition
331I.
331J. Lemma
331K. Theorem
331L. Theorem
331M. Homogeneous Boolean algebras
331N. Proposition
331O.
331X. Basic exercises
331Y. Further exercises
331 Notes and comments
332. Classification of localizable measure algebras
332A. Lemma
332B. Maharam\'s theorem
332C. Corollary
332D. The cellularity of a Boolean algebra
332E. Proposition
332F. Corollary
332G. Definitions
332H. Lemma
332I. Lemma
332J.
332K. Remarks
332L. Proposition
332M. Lemma
332N. Lemma
332O. Lemma
332P. Proposition
332Q. Proposition
332R.
332S. Theorem
332T. Proposition
332X. Basic exercises
332Y. Further exercises
332 Notes and comments
333. Closed subalgebras
333A. Definitions
333B.
333C. Theorem
333D. Corollary
333E. Theorem
333F. Corollary
333G. Corollary
333H.
333I. Remarks
333J. Lemma
333K. Theorem
333L. Remark
333M. Lemma
333N. A canonical form for closed subalgebras
333O. Remark
333P.
333Q. Corollary
333R.
333X. Basic exercises
333Y. Further exercises
333 Notes and comments
334. Products
334A. Theorem
334B. Corollary
334C. Theorem
334D. Corollary
334E.
334X .Basic exercises
334Y. Further exercises
334 Notes and comments
Chapter 34. The lifting theorem
341. The lifting theorem
341A. Definition
341B. Remarks
341C. Definition
341D. Remarks
341E. Example
341F.
341G. Lemma
341H.
341I.
341J. Proposition
341K. The Lifting Theorem
341L. Remarks
341M.
341N. Extension of partial liftings
341O. Liftings and Stone spaces
341P. Proposition
341Q. Corollary
341X. Basic exercises
341Y. Further exercises
341Z. Problems
341 Notes and comments
342. Compact measure spaces
342A. Definitions
342B.
342C. Corollary
342D. Lemma
342E. Corollary
342F. Corollary
342G.
342H. Proposition
342I. Proposition
342J. Examples
342K.
342L. Theorem
342M.
*342N. Example
342X. Basic exercises
342Y. Further exercises
342 Notes and comments
343. Realization of homomorphisms
343A. Preliminary remarks
343B. Theorem
343C. Examples
343D. Uniqueness of realizations
343E. Lemma
343F. Proposition
343G. Corollary
343H. Examples
343I. Example
343J. The split interval
343K.
343L.
343M. Example
343X. Basic exercises
343Y. Further exercises
343 Notes and comments
344. Realization of automorphisms
344A. Stone spaces
344B. Theorem
344C. Corollary
344D.
344E. Theorem
344F. Corollary
344G. Corollary
344H. Lemma
344I. Theorem
344J. Corollary
344K. Corollary
344L.
344X. Basic exercises
344Y. Further exercises
344 Notes and comments
345. Translation-invariant liftings
345A. Translation-invariant liftings
345B. Theorem
345C. Theorem
345D.
345E.
345F. Proposition
345X. Basic exercises
345Y. Further exercises
345 Notes and comments
346. Consistent liftings
346A. Definition
346B. Lemma
346C. Theorem
346D.
346E. Theorem
346F.
346G. Theorem
346H. Theorem
346I. Theorem
346J. Consistent liftings
346K. Lemma
346L. Proposition
346X. Basic exercises
346Y. Further exercises
346Z. Problems
346 Notes and comments
Concordance
Measure Theory 3-2_Measure Algebras(2e,2012,469p)D.H.Fremlin_9780956607119
Contents
Chapter 35. Riesz spaces
351. Partially ordered linear spaces
351A. Definition
351B. Elementary facts
351C. Positive cones
351D. Suprema and infima
351E. Linear subspaces
351F. Positive linear operators
351G. Order-continuous positive linear operators
351H. Riesz homomorphisms
351I. Solid sets
351J. Proposition
351K. Lemma
351L. Products
351M. Reduced powers of R
351N.
351O. Lemma
351P. Lemma
351Q.
351R. Archimedean spaces
351X. Basic exercises
351Y. Further exercises
351 Notes and comments
352. Riesz spaces
352A.
352B. Lemma
352C. Notation
352D. Elementary identities
352E. Distributive laws
352F. Further identities and inequalities
352G. Riesz homomorphisms
352H. Proposition
352I. Riesz subspaces
352J. Solid subsets
352K. Products
352L. Theorem
352M. Corollary
352N. Order-density and order-continuity
352O. Bands
352P. Complemented bands
352Q. Theorem
352R. Projection bands
352S. Proposition
352T. Products again
352U. Quotient spaces
352V. Principal bands
352W. f-algebras
352X. Basic exercises
352 Notes and comments
353. Archimedean and Dedekind complete Riesz spaces
353A. Proposition
353B. Proposition
353C. Corollary
353D. Proposition
353E. Lemma
353F. Lemma
353G. Dedekind completeness
353H. Proposition
353I. Proposition
353J. Proposition
353K. Proposition
353L. Order units
353M. Theorem
353N. Lemma
353O. f-algebras
353P. Proposition
353Q. Proposition
353X. Basic exercises
353Y. Further exercises
353 Notes and comments
354. Banach lattices
354A. Definitions
354B. Lemma
354C. Lemma
354D.
354E. Proposition
354F. Lemma
354G. Definitions
354H. Examples
354I. Lemma
354J. Proposition
354K. Theorem
354L. Corollary
354M.
354N. Theorem
354O. Proposition
354P. Uniform integrability in L-spaces
354Q.
354R.
354X. Basic exercises
354Y. Further exercises
354 Notes and comments
355. Spaces of linear operators
355A. Definition
355B. Lemma
355C. Theorem
355D. Lemma
355E. Theorem
355F. Theorem
355G. Definition
355H. Theorem
355I. Theorem
355J. Proposition
355K. Proposition
355X. Basic exercises
355Y. Further exercises
355 Notes and comments
356. Dual spaces
356A. Definition
356B. Theorem
356C. Proposition
356D. Proposition
356E. Biduals
356F. Theorem
356G. Lemma
356H. Lemma
356I. Theorem
356J. Definition
356K. Proposition
356L. Proposition
356M. Proposition
356N. L- and M-spaces
356O. Theorem
356P. Proposition
356Q. Theorem
356X. Basic exercises
356Y. Further exercises
356 Notes and comments
Chapter 36. Function Spaces
361. S
361A. Boolean rings
361B. Definition
361C. Elementary facts
361D. Construction
361E.
361F.
361G. Theorem
361H. Theorem
361I. Theorem
361J.
361K. Proposition
361L. Proposition
361M. Proposition
361X. Basic exercises
361Y. Further exercises
361 Notes and comments
362. S~
362A. Theorem
362B. Spaces of finitely additive functionals
362C.
362D.
362E. Uniformly integrable sets
362X. Basic exercises
362Y. Further exercises
362 Notes and comments
363. L^∞
363A. Definition
363B. Theorem
363C. Proposition
363D. Proposition
363E. Theorem
363F. Theorem
363G. Corollary
363H. Representations of L^\\infty ( \\mathfrak{A} )
363I. Corollary
363J. Recovering the algebra \\mathfrak{A}
363K. Dual spaces of L^\\infty
*363L. Integration with respect to a finitely additive functional
363M.
363N.
363O. Corollary
363P. Corollary
363Q.
363R.
363S. The Banach-Ulam problem
363X. Basic exercises
363Y. Further exercises
363 Notes and comments
364. L^0
364A. The set L^0( \\mathfrak{A} )
364B. Proposition
364C. Theorem
364D. Theorem
364E.
364F.
364G. Definition
364H. Proposition
364I. Examples
364J. Embedding S and L^\\infty in L^0
364K. Corollary
364L. Suprema and infima in L^0
364M.
364N. The multiplication of L^0
364O. Recovering the algebra
364P.
364Q. Proposition
364R. Products
*364S. Regular open algebras
*364T. Theorem
*364U. Compact spaces
*364V. Theorem
364X. Basic exercises
364Y. Further exercises
364 Notes and comments
365. L^1
365A. Definition
365B. Theorem
365C.
365D. Integration
365E. The Radon-Nikodým theorem again
365F.
365G. Semi-finite algebras
365H. Measurable transformations
365I. Theorem
365J. Corollary
365K. Theorem
365L. The duality between L^1 and L^\\infty
365M. Theorem
365N. Corollary
365O. Theorem
365P. Theorem
365Q. Proposition
365R. Conditional expectations
365S. Recovering the algebra: Proposition
365T.
365U. Uniform integrability
365X. Basic exercises
365Y. Further exercises
365 Notes and comments
366. L^p
366A. Definition
366B. Theorem
366C. Corollary
366D.
366E. Proposition
366F.
366G. Lemma
366H. Theorem
366I. Corollary
366J. Corollary
366K. Corollary
366L. Corollary
*366M. Complex L^p spaces
366X. Basic exercises
366Y. Further exercises
366 Notes and comments
367. Convergence in measure
367A. Order*-convergence
367B. Lemma
367C. Proposition
367D.
367E.
367F.
367G. Corollary
367H. Proposition
367I. Dominated convergence
367J. The Martingale Theorem
367K.
367L.
367M. Theorem
367N. Proposition
367O. Theorem
367P. Proposition
367Q.
367R.
367S. Proposition
367T. Intrinsic description of convergence in measure
*367U. Theorem
*367V. Corollary
*367W. Independence
367X. Basic exercises
367Y. Further exercises
367 Notes and comments
368. Embedding Riesz spaces in L^0
368A. Lemma
368B. Theorem
368C. Corollary
368D. Corollary
368E. Theorem
368F. Corollary
368G. Corollary
368H. Corollary
368I. Corollary
368J. Definition
368K.
368L. Definition
368M. Theorem
368N Weakly (σ, ∞)-distributive Riesz spaces
368O. Lemma
368P. Proposition
368Q. Theorem
368R. Corollary
368S. Corollary
368X. Basic exercises
368Y. Further exercises
368 Notes and comments
369. Banach function spaces
369A. Theorem
369B. Corollary
369C.
369D. Corollary
369E. Kakutani\'s theorem
369F.
369G. Proposition
369H. Associate norms
369I. Theorem
369J. Theorem
369K. Corollary
369L. L^p
369M. Proposition
369N.
369O. Proposition
369P.
369Q. Corollary
369R.
369X. Basic exercises
369Y. Further exercises
369 Notes and comments
Chapter 37. Linear operators between function spaces
371. The Chacon-Krengel theorem
371A. Lemma
371B. Theorem
371C. Theorem
371D. Corollary
371E. Remarks
371F. The class T^{(0)}
371G. Proposition
371H. Remark
371X. Basic exercises
371Y. Further exercises
371 Notes and comments
372. The ergodic theorem
372A. Lemma
372B. Lemma
372C. Maximal Ergodic Theorem
372D.
372E. Corollary
372F. The Ergodic Theorem: second form
372G. Corollary
372H.
372I.
372J. The Ergodic Theorem: third form
372K. Remark
372L. Continued fractions
372M. Theorem
372N. Corollary
372O. Mixing and ergodic transformations
372P.
372Q.
372R. Remarks
372S.
372X. Basic exercises
372Y. Further exercises
372 Notes and comments
373. Decreasing rearrangements
373A. Definition
373B. Proposition
373C. Decreasing rearrangements
373D. Lemma
373E. Theorem
373F. Theorem
373G. Lemma
373H. Lemma
373I. Lemma
373J. Corollary
373K. The very weak operator topology of T
373L. Theorem
373M. Corollary
373N. Corollary
373O. Theorem
373P. Theorem
373Q. Corollary
373R. Order-continuous operators: Proposition
373S. Adjoints in T^{(0)}
373T. Corollary
373U. Corollary
373X. Basic exercises
373Y. Further exercises
373 Notes and comments
374. Rearrangement-invariant spaces
374A. T-invariance
374B.
374C.
374D.
374E.
374F. Remarks
374G. Definition
374H. Proposition
374I. Corollary
374J. Lemma
374K. Theorem
374L. Lemma
374M. Proposition
374X. Basic exercises
374Y. Further exercises
374 Notes and comments
375. Kwapien\'s theorem
375A. Theorem
375B. Proposition
375C. Theorem
375D. Corollary
375E. Theorem
375F.
375G. Lemma
375H. Lemma
375I. Lemma
375J. Theorem
375K. Corollary
375L. Corollary
375X. Basic exercises
375Y. Further exercises
375Z. Problem
375 Notes and comments
376. Kernel operators
376A. Kernel operators
376B. The canonical map L^0 × L^0 -> L^0
376C.
376D. Abstract integral operators
376E. Theorem
376F. Corollary
376G. Lemma
376H. Theorem
376I.
376J. Corollary
376K.
376L. Lemma
376M. Theorem
376N. Corollary: Dunford\'s theorem
376O.
376P. Theorem
376Q. Corollary
376R.
376S. Theorem
376X. Basic exercises
376Y. Further exercises
376 Notes and comments
*377 .Function spaces of reduced products
377A. Proposition
377B. Theorem
377C. Theorem
377D.
377E. Proposition
377F.
377G. Projective limits
377H. Inductive limits
377X. Basic exercises
377Y. Further exercises
377 Notes and comments
Chapter 38. Automorphism groups
381. Automorphisms of Boolean algebras
381A. The group Aut \\mathfrak{A}
381B.
381C.
381D. Corollary
381E. Lemma
381F. Corollary
381G. Corollary
381H. Proposition
381I. Full and countably full subgroups
381J. Lemma
381K. Lemma
381L. Lemma
381M.
381N. Lemma
381O. Lemma
381P. Proposition
381Q.
381R. Cyclic automorphisms
381S. Lemma
381X. Basic exercises
381Y. Further exercises
382. Factorization of automorphisms
382A. Definitions
382B. Lemma
382C. Corollary
382D. Lemma
382E. Corollary
382F. Corollary
382G. Lemma
382H. Lemma
382I. Lemma
382J. Lemma
382K. Lemma
382L. Lemma
382M. Theorem
382N. Corollary
382O. Definition
382P. Lemma
382Q. Lemma
382R. Theorem
382S. Corollary
382X. Basic exercises
382Y. Further exercises
382 Notes and comments
383. Automorphism groups of measure algebras
383A. Definition
383B. Lemma
383C. Corollary
383D. Theorem
383E. Lemma
383F. Lemma
383G. Lemma
383H. Corollary
383I. Normal subgroups of Aut \\mathfrak{A} and Aut_{\\bar{\\mu}} \\mathfrak{A}
383J.
383K.
383L. Corollary
383X. Basic exercises
383Y. Further exercises
383 Notes and comments
384. Outer automorphisms
384A. Lemma
384B. A note on supports
384C. Lemma
384D. Theorem
384E.
384F. Corollary
384G. Corollary
384H. Definitions
384I. Lemma
384J. Theorem
384K. Corollary
384L. Examples
384M. Theorem
384N.
384O. Corollary
384P.
384Q. Example
384X. Basic exercises
384Y. Further exercises
384 Notes and comments
385. Entropy
385A. Notation
385B. Lemma
385C. Definition
385D. Definition
385E. Elementary remarks
385F. Definition
385G. Lemma
385H. Corollary
385I. Lemma
385J. Lemma
385K. Definition
385L. Lemma
385M. Definition
385N. Lemma
385O. Lemma
385P. Theorem (Kolmogorov 58, Sinai 59)
385Q. Bernoulli shifts
385R. Theorem
385S. Remarks
385T. Isomorphic homomorphisms
385U. Definition
385V.
385X. Basic exercises
385Y. Further exercises
385 Notes and comments
386. More about entropy
386A.
386B. Corollary
386C. The Halmos-Rokhlin-Kakutani lemma
386D. Corollary
386E.
386F. Corollary
386G. Definition
386H. Lemma
386I. Corollary
386J.
386K. Lemma
386L. Lemma
386M. Lemma
386N. Lemma
386O. Lemma
386X. Basic exercises
386Y. Further exercises
386 Notes and comments
387. Ornstein\'s theorem
387A.
387B. Remarks
387C. Lemma
387D. Corollary
387E. Sinaĭ\'s theorem (atomic case) (SINAĬ 62)
387F. Lemma
387G. Lemma
387H. Lemma
387I. Ornstein\'s theorem (finite entropy case)
387J.
387K. Ornstein\'s theorem (infinite entropy case)
387L. Corollary: Sinaĭ\'s theorem (general case)
387X. Basic exercises
387Y. Further exercises
387 Notes and comments
388. Dye\'s theorem
388A. Orbit structures
388B. Corollary
388C.
388D. von Neumann automorphisms
388E. Example
388F.
388G. Lemma
388H. Lemma
388I. Lemma
388J. Lemma
388K. Theorem
388L. Theorem
388X. Basic exercises
388Y. Further exercises
388 Notes and comments
Chapter 39. Measurable algebras
391. Kelley\'s theorem
391A. Proposition
391B. Definition
391C. Proposition
391D. Theorem (Kantorovich Vulikh & Pinsker 50)
391E.
391F. Theorem
391G. Corollary
391H. Definition
391I. Proposition
391J. Theorem
391K. Corollary
391X. Basic exercises
391Y. Further exercises
391 Notes and comments
392. Submeasures
392A. Definition
392B.
392C. Proposition
392D. Lemma
392E. Lemma
392F. Theorem
392G. Corollary
392H.
392I. Corollary
392J. Proposition
*392K. Products of submeasures
392X. Basic exercises
392Y. Further exercises
392 Notes and comments
393. Maharam submeasures
393A. Definition
393B. Lemma
393C. Proposition
393D. Theorem
393E. Maharam algebras
393F. Lemma
393G. Proposition
393H. Proposition
393I. Proposition
393J. Lemma
*393K. Theorem
393L.
393M. Lemma
393N. Proposition
393O. Proposition
393P. Lemma
393Q. Theorem (Balcar Głowczynski & Jech 98, Balcar Jech & Pazák 05)
393R. Definition
393S. Theorem (TODORČEVIĆ 04)
393X. Basic exercises
393Y. Further exercises
393 Notes and comments
394. Talagrand\'s example
394A.
394B. Lemma
394C. Definitions
394D. Very elementary facts
394E. Lemma
394F. Corollary
394G.
394H. Definitions
394I. Proposition
394J. Lemma
394K. Lemma
394L. Lemma
394M. Theorem
394N. Remarks
*394O. Control measures
*394P. Example
*394Q.
394X. Basic exercises
394Y. Further exercises
394Z. Problems
394 Notes and comments
395. Kawada\'s theorem
395A. Definitions
395B.
395C. Lemma
395D. Theorem
395E. Definition
395F. Proposition
395G. The fixed-point subalgebra of a group
395H.
395I.
395J. Notation
395K. Lemma
395L. Lemma
395M. Lemma
395N.
395O.
395P. Theorem
395Q. Corollary: Kawada\'s theorem
395R.
395X. Basic exercises
395Y. Further exercises
395Z. Problem
395 Notes and comments
396. The Hajian-Ito theorem
396A. Lemma
396B. Theorem (Hajian & Ito 69)
396C. Remark
396X. Basic exercises
396Y. Further exercises
396 Notes and comments
Appendix to Volume 3 - Useful Facts
3A1. Set Theory
3A1A. The axioms of set theory
3A1B. Definition
3A1C. Calculation of cardinalities
3A1D. Cardinal exponentiation
3A1E. Definition
3A1F. Cofinal sets
3A1G. Zorn\'s Lemma
3A1H. Natural numbers and finite ordinals
3A1I. Definitions
3A1J. Subsets of given size
3A1K.
3A2. Rings
3A2A. Definition
3A2B. Elementary facts
3A2C. Subrings
3A2D. Homomorphisms
3A2E. Ideals
3A2F. Quotient rings
3A2G. Factoring homomorphisms through quotient rings
3A2H. Product rings
3A3. General topology
3A3A. Taxonomy of topological spaces
3A3B. Elementary relationships
3A3C. Continuous functions
3A3D. Compact spaces
3A3E. Dense sets
3A3F. Meager sets
3A3G. Baire\'s theorem for locally compact Hausdorff spaces
3A3H. Corollary
3A3I. Product spaces
3A3J. Tychonoff\'s theorem
3A3K. The spaces {0, 1}^{I}
3A3L. Cluster points of filters
3A3M. Topology bases
3A3N. Uniform convergence
3A3O. One-point compactifications
3A3P. Topologies defined from a sequential convergence
3A3Q. Miscellaneous definitions
3A4. Uniformities
3A4A. Uniformities
3A4B. Uniformities and pseudometrics
3A4C. Uniform continuity
3A4D. Subspaces
3A4E. Product uniformities
3A4F. Completeness
3A4G. Extension of uniformly continuous functions
3A4H. Completions
3A4I. A note on metric spaces
3A5. Normed spaces
3A5A. The Hahn-Banach theorem
3A5B. Cones
3A5C. Hahn-Banach theorem: geometric forms
3A5D. Separation from finitely-generated cones
3A5E. Weak topologies
3A5F. Weak* topologies
3A5G. Reflexive spaces
3A5H. Uniform Boundedness Theorem
*3A5I. Strong operator topologies
3A5J. Completions
3A5K. Normed algebras
3A5L. Compact operators
3A5M. Hilbert spaces
*3A5N. Bounded sets in linear topological spaces
3A6. Group Theory
3A6A. Definition
3A6B. Definition
3A6C. Normal subgroups
Concordance
References for Volume 3
Index to volumes 1, 2 and 3
Principal topics and results
General index
Measure Theory 4-1_Topological Measure Spaces(2e,2013,577p)D.H.Fremlin_9780956607126
Contents
General introduction
Introduction to Volume 4
Note on second printing
Note on second (`Lulu\') edition
Chapter 41. Topologies and Measures I
411. Definitions
411A.
411B.
411C. Definition
411D.
411E.
411F.
411G. Elementary facts
411H.
411I. Remarks
411J.
411K. Borel and Baire measures
411L.
411M. Definition
411N.
411O. Example
411P. Example: Stone spaces
411Q. Example: Dieudonné\'s measure
411R. Example: The Baire σ-algebra of ω_1
411X. Basic exercises
411Y. Further exercises
411 Notes and comments
412. Inner regularity
412A.
412B. Corollary
412C.
412D.
412E. Theorem
412F. Lemma
412G. Theorem
412H. Proposition
412I. Lemma
412J. Proposition
412K. Proposition
412L. Corollary
412M. Corollary
412N. Lemma
412O. Lemma
412P. Proposition
412Q. Proposition
412R. Lemma
412S. Proposition
412T. Lemma
412U. Proposition
412V. Corollary
*412W. Outer regularity
412X. Basic exercises
412Y. Further exercises
412 Notes and comments
413. Inner measure constructions
413A.
413B.
413C. Measures from inner measures
413D. The inner measure defined by a measure
413E.
413F.
413G.
413H.
413I. Theorem (Topsøe 70A)
413J. Theorem
413K. Corollary
413L.
413M. Corollary
413N.
413O. Corollary
413P.
413Q. Theorem
413R.
413S. Corollary
413T.
413X. Basic exercises
413Y. Further exercises
413 Notes and comments
414. τ-additivity
414A. Theorem
414B. Corollary
414C. Corollary
414D. Corollary
414E. Corollary
414F. Corollary
414G. Corollary
414H. Corollary
414I. Proposition
414J. Theorem
414K. Proposition
414L. Lemma
414M. Proposition
414N. Proposition
414O.
414P. Density topologies
414Q. Lifting topologies
414R. Proposition
414X. Basic exercises
414Y. Further exercises
414 Notes and comments
415. Quasi-Radon measure spaces
415A. Theorem
415B. Theorem
415C.
415D.
415E.
415F. Corollary
415G. Comparing quasi-Radon measures
415H. Uniqueness of quasi-Radon measures
415I. Proposition
415J. Proposition
415K.
415L. Proposition
415M. Corollary
415N. Corollary
415O. Proposition
415P. Proposition
415Q.
415R. Proposition
415X. Basic exercises
415Y. Further exercises
415 Notes and comments
416. Radon measure spaces
416A. Proposition
416B. Corollary
416C.
416D.
416E. Specification of Radon measures
416F. Proposition
416G.
416H. Corollary
416I.
416J.
416K. Proposition (see TOPSØE 70A)
416L. Proposition
416M. Corollary
416N. Henry\'s theorem (Henry 69)
416O. Theorem
416P. Theorem
416Q. Proposition
416R. Theorem
416S.
416T.
416U. Theorem
416V. Stone spaces
416W. Compact measure spaces
416X. Basic exercises
416Y. Further exercises
416 Notes and comments
417. τ-additive product measures
417A. Lemma
417B. Lemma
417C. Theorem (RESSEL 77)
417D. Multiple products
417E. Theorem
417F. Corollary
417G. Notation
417H. Fubini\'s theorem for τ-additive product measures
417I.
417J.
417K. Proposition
417L. Corollary
417M. Proposition
417N. Theorem
417O. Theorem
417P. Theorem
417Q. Theorem
417R. Notation
417S.
417T. Proposition
417U. Proposition
417V. Proposition
417X. Basic exercises
417Y. Further exercises
417 Notes and comments
418. Measurable functions and almost continuous functions
418A. Proposition
418B. Proposition
418C. Proposition
418D. Proposition
418E. Theorem
418F. Proposition
418G. Proposition
418H. Proposition
418I.
418J. Theorem
418K. Corollary
418L.
418M. Prokhorov\'s theorem
418N. Remarks
418O.
418P. Proposition
418Q. Corollary
418R.
418S. Corollary
418T. Corollary (MAULDIN & STONE 81)
*418U. Independent families of measurable functions
418X. Basic exercises
418Y. Further exercises
418 Notes and comments
419. Examples
419A. Example
419B. Lemma
419C. Example (FREMLIN 75B)
419D. Example (FREMLIN 75B)
419E. Example (FREMLIN 76)
419F. Theorem (RAO 69)
419G. Corollary (ULAM 30)
419I.
419J. Example
419K. Example (BLACKWELL 56)
419L. The split interval again
419X. Basic exercises
419Y. Further exercises
419 Notes and comments
Chapter 42. Descriptive set theory
421. Souslin\'s operation
421A. Notation
421B. Definition
421C. Elementary facts
421D.
421E. Corollary
421F. Corollary
421G. Proposition
421H.
421I.
421J. Proposition
421K. Definition
421L. Proposition
421M. Proposition
*421N.
*421O. Theorem
*421P. Corollary
*421Q. Lemma
421X. Basic exercises
421Y. Further exercises
421 Notes and comments
422. K-analytic spaces
422A. Definition
422B.
422C. Proposition
422D. Lemma
422E.
422F Definition (FROLÍK 61)
422G. Theorem
422H. Theorem
422I.
422J. Corollary
*422K.
422X. Basic exercises
422Y. Further exercises
422 Notes and comments
423. Analytic spaces
423A. Definition
423B. Proposition
423C. Theorem
423D. Corollary
423E. Theorem
423F. Proposition
423G. Lemma
423H. Lemma
423I. Theorem
423J. Lemma
423K. Corollary
423L. Proposition
423M.
423N.
423O. Corollary
*423P. Constituents of coanalytic sets
*423Q. Remarks
*423R. Coanalytic and PCA sets
423S. Proposition
423X. Basic exercises
423Y. Further exercises
423 Notes and comments
424. Standard Borel spaces
424A. Definition
424B. Proposition
424C. Theorem
424D. Corollary
424E. Proposition
424F. Corollary
424G. Proposition
*424H.
424X. Basic exercises
424Y. Further exercises
424 Notes and comments
*425. Realization of automorphisms
425A.
425B. Lemma
425C. Master actions
425D. Törnquist\'s theorem (TÖRNQUIST 11)
425E. Scholium
425X. Basic exercises
425Y. Further exercises
425Z. Problems
425 Notes and comments
Chapter 43. Topologies and measures II
431. Souslin\'s operation
431A. Theorem
431B. Corollary
431C. Corollary
431D. Theorem
431E. Corollary
*431F.
*431G.
431X. Basic exercises
431Y. Further exercises
431 Notes and comments
432. K-analytic spaces
432A. Proposition
432B. Theorem
432C. Proposition
432D. Theorem (ALDAZ & RENDER 00)
432E. Corollary
432F. Corollary
432G. Corollary
432H. Corollary
432I. Corollary
432J. Capacitability
432K. Theorem (CHOQUET 55)
432L. Proposition
432X. Basic exercises
432Y. Further exercises
432 Notes and comments
433. Analytic spaces
433A. Proposition
433B. Lemma
433C. Theorem
433D. Theorem
433E. Proposition
433F.
433G. Proposition
433H. Proposition
433I.
433J. Proposition
433K.
433L. Proposition
433X. Basic exercises
433Y. Further exercises
433 Notes and comments
434. Borel measures
434A. Types of Borel measures
434B. Compact, analytic and K-analytic spaces
434C. Radon spaces
434D. Universally measurable sets
434E. Universally Radon-measurable sets
434F. Elementary properties of Radon spaces
434G.
434H. Proposition
434I. Proposition
434J. Proposition
434K.
434L.
434M.
434N. Proposition
434O. Quasi-dyadic spaces
434P. Proposition
434Q. Theorem (FREMLIN & GREKAS 95)
434R.
*434S.
*434T.
434X. Basic exercises
434Y. Further exercises
434Z. Problems
434 Notes and comments
435. Baire measures
435A. Types of Baire measures
435B. Theorem
435C. Theorem (MARÍK 57)
435D.
435E.
435F. Elementary facts
435G. Proposition
435H. Corollary
435X. Basic exercises
435Y. Further exercises
435 Notes and comments
436. Representation of linear functionals
436A. Definition
436B. Definition
436C. Lemma
436D. Theorem
436E. Proposition
436F.
436G. Definition
436H. Theorem
436I. Lemma
436J. Riesz Representation Theorem (first form)
436K. Riesz Representation Theorem (second form)
*436L.
*436M. Corollary
436X. Basic exercises
436Y. Further exercises
436 Notes and comments
437. Spaces of measures
437A. Smooth and sequentially smooth duals
437B. Signed measures
437C. Theorem
437D. Remarks
437E. Corollary
437F. Proposition
437G. Definitions
437H. Theorem
437I. Proposition
437J. Vague and narrow topologies
437K. Proposition
437L. Corollary
437M. Theorem (RESSEL 77)
437N.
437O. Uniform tightness
437P. Proposition
437Q. Two metrics
437R. Theorem
437S.
437T.
437U.
437V. Theorem
437X. Basic exercises
437Y. Further exercises
437 Notes and comments
438. Measure-free cardinals
438A. Measure-free cardinals
438B.
438C.
438D.
438E. Proposition
438F. Proposition
438G. Corollary
438H.
438I. Proposition
438J.
438K. Hereditarily weakly θ-refinable spaces
438L. Lemma
438M. Proposition (GARDNER 75)
438N.
438O. Lemma
438P. Lemma
438Q. Theorem
438R. Corollary
*438S. Càllàl functions
438T. Proposition
438U.
438X. Basic exercises
438Y. Further exercises
438 Notes and comments
439. Examples
439A. Example
439B. Definition
439C. Proposition
439D. Remarks
439E. Lemma
439F. Proposition
439G. Corollary
439H. Corollary
439I. Example
439J. Example
439K. Example
439L. Example
439M. Example
439N. Example
439O.
439P. Example (cf. MORAN 68)
439Q. Example
439R. Example
439S.
439X. Basic exercises
439Y. Further exercises
439 Notes and comments
Chapter 44. Topological groups
441. Invariant measures on locally compact spaces
441A. Group actions
441B.
441C. Theorem (STEINLAGE 75)
441D.
441E. Theorem
441F.
441G. The topology of an isometry group
441H. Theorem
441I. Remarks
441J.
441K. Theorem
1L P.roposition
441X. Basic exercises
441Y. Further exercises
441 Notes and comments
442. Uniqueness of Haar measures
442A. Lemma
442B. Theorem
442C. Proposition
442D. Remark
442E. Lemma
442F. Domains of Haar measures
442G. Corollary
442H. Remark
442I. The modular function
442J. Proposition
442K. Theorem
442L. Corollary
442X. Basic exercises
442Y. Further exercises
442Z. Problem
442 Notes and comments
443. Further properties of Haar measure
443A. Haar measurability
443B. Lemma
443C. Theorem
443D. Proposition
443E. Corollary
443F.
443G.
443H. Theorem
443I. Corollary
443J. Proposition
443K. Theorem
443L. Corollary
443M. Theorem (HALMOS 50)
443N.
443O.
443P. Quotient spaces
443Q. Theorem
443R. Theorem
443S. Applications
443T. Theorem
443U. Transitive actions
443X. Basic exercises
443Y. Further exercises
443 Notes and comments
444. Convolutions
444A. Convolution of measures
444B. Proposition
444C. Theorem
444D. Proposition
444E. The Banach algebra of τ-additive measures
444F.
444G. Corollary
444H. Convolutions of measures and functions
444I. Proposition
444J. Convolutions of functions and measures
444K. Proposition
444L. Corollary
444M. Proposition
444N.
444O. Convolutions of functions
444P. Proposition
444Q. Proposition
444R. Proposition
444S. Remarks
444T. Proposition
444U. Corollary
444V.
444X. Basic exercises
444Y .Further exercises
444 Notes and comments
445. The duality theorem
445A. Dual groups
445B. Examples
445C. Fourier-Stieltjes transforms
445D. Theorem
445E.
445F. Fourier transforms of functions
445G. Proposition
445H. Theorem
445I. The topology of the dual group
445J. Corollary
445K. Proposition
445L. Positive definite functions
445M. Proposition
445N. Bochner\'s theorem (HERGLOTZ 1911, BOCHNER 33, WEIL 40)
445O. Proposition
445P. The Inversion Theorem
445Q. Remark
445R. The Plancherel Theorem
445S.
445T. Corollary
445U. The Duality Theorem (PONTRYAGIN 34, KAMPEN 35)
445X. Basic exercises
445Y. Further exercises
445 Notes and comments
446. The structure of locally compact groups
446A. Finite-dimensional representations
446B. Theorem
446C. Corollary
*446D. Notation
*446E. Lemma
*446F. Lemma
*446G. `Groups with no small subgroups\'
*446H. Lemma
*446I. Lemma
*446J. Lemma
*446K. Lemma
*446L. Definition
*446M. Proposition
*446N. Proposition
*446O. Theorem
*446P. Corollary
446X. Basic exercises
446Y. Further exercises
446 Notes and comments
447. Translation-invariant liftings
447A. Liftings and lower densities
447B. Lemma
447C. Vitali\'s theorem
447D. Theorem
447E.
447F. Lemma
447G. Lemma
447H. Lemma
447I. Theorem (IONESCU TULCEA & IONESCU TULCEA 67)
447J. Corollary
447X. Basic exercises
447Y. Further exercises
447 Notes and comments
448. Polish group actions
448A. Definitions
448B.
448C. Lemma
448D. Theorem
448E. Definition
448F.
448G.
448H. Lemma
448I. Notation
448K.
448L.
448M. Lemma
448N. Theorem
448O.
448P.
448Q.
448R. Lemma
448S. Mackey\'s theorem (MACKEY 62)
448T. Corollary
448X. Basic exercises
448Y. Further exercises
448 Notes and comments
449. Amenable groups
449A. Definition
449B. Lemma
449C. Theorem
449D. Theorem
449E. Corollary
449F. Corollary
449G. Example
449H.
449I. Notation
449J. Theorem
449K. Proposition
449L.
449M. Corollary
449N. Theorem
449O. Corollary (BANACH 1923)
449X. Basic exercises
449Y. Further exercises
449 Notes and comments
Chapter 45. Perfect measures and disintegrations
451. Perfect, compact and countably compact measures
451A.
451B.
451C. Proposition (RYLL-NARDZEWSKI 53)
451D. Proposition
451E. Proposition
451F. Lemma (SAZONOV 66)
451G. Proposition
451H. Lemma
451I. Theorem
451J. Theorem
451K.
*451L.
451M.
451N. Proposition
451O. Corollary
451P. Corollary
451Q.
451R. Lemma
451S. Proposition
451T. Theorem (FREMLIN 81, KOUMOULLIS & PRIKRY 83)
451U. Example (VINOKUROV & MAKHKAMOV 73, MUSIAŁ 76)
*451V Weakly α-favourable spaces
451X. Basic exercises
451Y. Further exercises
451 Notes and comments
452. Integration and disintegration of measures
452A. Lemma
452B. Theorem
452C. Theorem
452D. Theorem
452E.
452F. Proposition
452G.
452H. Lemma
452I. Theorem (PACHL 78)
452J. Remarks
452K. Example
452L.
452M.
452N. Corollary
452O. Proposition
452P. Corollary (cf. BLACKWELL 56)
452Q. Disintegrations and conditional expectations
*452R.
*452S. Corollary (PACHL 78)
452T.
452X. Basic exercises
452Y. Further exercises
452 Notes and comments
453. Strong liftings
453A.
453B.
453C. Proposition
453D. Proposition
453E. Proposition
453F. Proposition
453G. Corollary
453H. Lemma
453I. Proposition
453J. Corollary
453K.
453L. Remark
453M. Strong liftings and Stone spaces
453N. Losert\'s example (LOSERT 79)
453X. Basic exercises
453Y. Further exercises
453Z. Problems
453 Notes and comments
454. Measures on product spaces
454A. Theorem
454B. Corollary
454C. Theorem (MARCZEWSKI & RYLL-NARDZEWSKI 53)
454D. Theorem (KOLMOGOROV 33, §III.4)
454E. Corollary
454F. Corollary
454G. Corollary
454H. Corollary
454I. Remarks
454J. Distributions of random processes
454K. Definition
454L. Independence
454M.
454N.
454O. Proposition
454P. Theorem
454Q. Continuous processes
454R. Proposition
454S. Corollary
454T.
454X. Basic exercises
454Y. Further exercises
454 Notes and comments
455. Markov and Lévy processes
455A. Theorem
455B. Lemma
455C. Theorem
455D. Remarks
455E. Theorem
455F.
455G. Theorem
455H. Corollary
455I.
455J. Theorem
455K. Corollary
455L. Stopping times
455M. Hitting times
455N.
455O.
455P.
455Q. Lévy processes
455R. Theorem
455S. Lemma
455T. Corollary
455U. Theorem
455X. Basic exercises
455Y. Further exercises
455 Notes and comments
456. Gaussian distributions
456A. Definitions
456B.
456C.
456D. Gaussian processes
456E. Independence and correlation
456F. Proposition
456G.
456H. The support of a Gaussian distribution
456I. Remarks
456J. Universal Gaussian distributions
456K. Proposition
456L. Lemma
456M. Cluster sets
456N. Lemma
456O.
456P. Corollary
456Q. Proposition
456X. Basic exercises
456Y. Further exercises
456 Notes and comments
457. Simultaneous extension of measures
457A.
457B. Corollary
457C. Corollary
*457D.
457E. Proposition
457F. Proposition
457G. Theorem
457H. Example
457I. Example
457J. Example
457K.
457L. Theorem
457M.
457N. Remarks
457X. Basic exercises
457Y. Further exercises
457Z. Problems
457 Notes and comments
458. Relative independence and relative products
458A. Relative independence
458B.
458C. Proposition
458D. Proposition
458E. Example
458F.
*458G.
458H.
458I.
458J. Theorem
458K.
458L. Measure algebras
458M. Proposition
458N. Relative free products of probability algebras
458O. Theorem
458P.
458Q. Relative product measures
458R. Proposition
458S.
458T.
458U.
458X. Basic exercises
458Y. Further exercises
458 Notes and comments
459. Symmetric measures and exchangeable random variables
459A.
459B. Theorem
459C. Exchangeable random variables
459D.
459E.
459F. Lemma
459G. Lemma
459H. Theorem
459I.
459J. Corollary
459K.
459X. Basic exercises
459Y. Further exercises
459 Notes and comments
Concordance to chapters 41-45
References
Measure Theory 4-2_Topological Measure Spaces(2e,2013,573p)D.H.Fremlin_9780956607133
Contents
Chapter 46. Pointwise compact sets of measurable functions
461. Barycenters and Choquet\'s theorem
461A. Definitions
461B. Proposition
461C. Lemma
461D. Theorem
461E. Theorem
461F. Theorem
461G. Lemma
461H. Proposition
461I. Theorem
461J. Corollary: Kreĭn\'s theorem
461K. Lemma
461L. Lemma
461M. Theorem
461N. Lemma
461O. Lemma
461P. Theorem
461Q.
461R. Corollary
461X. Basic exercises
461Y. Further exercises
461 Notes and comments
462. Pointwise compact sets of continuous functions
462A. Definitions
*462B. Proposition (PRYCE 71)
*462C. Theorem (PRYCE 71)
*462D. Theorem
462E. Theorem
462F. Lemma
462G. Proposition
462H. Lemma
462I. Theorem
462J. Corollary
462K. Proposition
462L. Corollary
462X. Basic exercises
462Y. Further exercises
462Z. Problem
462 Notes and comments
463. \\mathfrak{T}_p and \\mathfrak{T}_m
463A. Preliminaries
463B. Lemma
463C. Proposition (IONESCU TULCEA 73)
463D. Lemma
463E. Proposition
463F. Corollary
463G. Theorem (IONESCU TULCEA 74)
463H. Corollary
463I. Lemma
463J. Lemma
463K. Fremlin\'s Alternative (FREMLIN 75A)
463L. Corollary
463M. Proposition
463N. Corollary
463X. Basic exercises
463Y. Further exercises
463Z. Problems
463 Notes and comments
464. Talagrand\'s measure
464A. The usual measure on PI
464B. Lemma
464C. Lemma
464D. Construction (TALAGRAND 80)
464E. Example
464F. The L-space \\ell^\\infty(I)^\\ast
464G.
464H.
464I. Measurable and purely non-measurable functionals
464J. Examples
464K. The space M_m
464L. The space M_{pnm}
464M. Theorem (FREMLIN & TALAGRAND 79)
464N. Corollary (FREMLIN & TALAGRAND 79)
464O. Remark
464P. More on purely non-measurable functionals
464Q. More on measurable functionals
464R. A note on \\ell^\\infty(I)
464X. Basic exercises
464Y. Further exercises
464Z. Problem
464 Notes and comments
465. Stable sets
465A. Notation
465B. Definition
465C.
465D.
465E. The topology \\mathfrak{T}_s(\\mathcal{L}^2, \\mathcal{L}^2)
465F. Lemma
465G. Theorem
465H.
465I.
465J.
465K. Lemma
465L. Lemma (TALAGRAND 87)
465M. Theorem (TALAGRAND 82, TALAGRAND 87)
465N. Theorem
465O. Stable sets in L^0
465P. Theorem
465Q. Remarks
465R. Theorem (TALAGRAND 84)
*465S. R-stable sets
*465T. Proposition (TALAGRAND 84)
*465U.
*465V. Remark
465X. Basic exercises
465Y. Further exercises
465 Notes and comments
466. Measures on linear topological spaces
466A. Theorem
466B. Corollary
466C. Definition
466D. Proposition (HANSELL 01)
466E. Corollary
466F. Proposition
466G. Definition
466H. Proposition (JAYNE & ROGERS 95)
466I. Examples
466J. Theorem
466K. Proposition
466L. Proposition
466M. Corollary
466N. Gaussian measures
466O. Proposition
466X. Basic exercises
466Y. Further exercises
466Z. Problems
466 Notes and comments
*467 .Locally uniformly rotund norms
467A. Definition
467B. Proposition
467C. A technical device
467D. Lemma
467E. Theorem
467F. Lemma
467G. Theorem
467H. Definitions
467I. Lemma
467J. Lemma
467K. Theorem
467L. Weakly compactly generated Banach spaces
467M. Proposition (TALAGRAND 75)
467N. Theorem
467O. Eberlein compacta
467P. Proposition
467X. Basic exercises
467Y. Further exercises
467 Notes and comments
Chapter 47. Geometric measure theory
471. Hausdorff measures
471A. Definition
471B. Definition
471C. Proposition
471D. Theorem
471E. Corollary
471F. Corollary
471G. Increasing Sets Lemma (DAVIES 70)
471H. Corollary
471I. Theorem
471J. Proposition
471K. Lemma
471L. Proposition
471M.
471N. Lemma
471O. Lemma
471P. Theorem
471Q.
471R. Lemma (HOWROYD 95)
471S. Theorem (HOWROYD 95)
471T. Proposition
471X. Basic exercises
471Y. Further exercises
471 Notes and comments
472. Besicovitch\'s Density Theorem
472A. Besicovitch\'s Covering Lemma
472B. Theorem
472C. Theorem
472D. Besicovitch\'s Density Theorem
*472E. Proposition
*472F. Theorem
472X. Basic exercises
472Y. Further exercises
472 Notes and comments
473. Poincaré\'s inequality
473A. Notation
473B. Differentiable functions
473C. Lipschitz functions
473D. Smoothing by convolution
473E. Lemma
473F. Lemma
473G. Lemma
473H. Gagliardo-Nirenberg-Sobolev inequality
473I. Lemma
473J. Lemma
473K. Poincaré\'s inequality for balls
473L. Corollary
473M. The case r = 1
473X. Basic exercises
473Y. Further exercises
473 Notes and comments
474. The distributional perimeter
474A. Notation
474B. The divergence of a vector field
474C. Invariance under isometries
474D. Locally finite perimeter
474E. Theorem
474F. Definitions
474G. The reduced boundary
474H. Invariance under isometries
474I. Half-spaces
474J. Lemma
474K. Lemma
474L .Two isoperimetric inequalities
474M. Lemma
474N. Lemma
474O. Definition
474P. Lemma
474Q. Lemma
474R. Theorem
474S. Corollary
474T. The Compactness Theorem
474X. Basic exercises
474Y. Further exercises
474 Notes and comments
475. The essential boundary
475A. Notation
475B. The essential boundary
475C. Lemma
475D. Lemma
475E. Lemma
475F. Lemma
475G. Theorem
475H. Proposition
475I. Lemma
475J. Lemma
475K. Lemma
475L. Theorem
475M. Corollary
475N. The Divergence Theorem
475O.
475P. Lemma
475Q. Theorem
475R. Convex sets in R^r
475S. Corollary: Cauchy\'s Perimeter Theorem
475T. Corollary: the Convex Isoperimetric Theorem
475X. Basic exercises
475Y. Further exercises
475 Notes and comments
476. Concentration of measure
476A. Proposition
476B. Lemma
476C. Proposition
476D. Concentration by partial reflection
476E. Lemma
476F. Theorem
476G. Theorem
476H. The Isoperimetric Theorem
476I. Spheres in inner product spaces
476J. Lemma
476K.
476L. Corollary
476X. Basic exercises
476Y. Further exercises
476 Notes and comments
477. Brownian motion
477A. Brownian motion
477B.
*477C.
477D. Multidimensional Brownian motion
477E. Invariant transformations of Wiener measure
477F. Proposition
477G. The strong Markov property
477H. Some families of σ-algebras
477I. Hitting times
477J.
477K. Typical Brownian paths
477L. Theorem
477X. Basic exercises
477Y. Further exercises
477 Notes and comments
478. Harmonic functions
478A. Notation
478B. Harmonic and superharmonic functions
478C. Elementary facts
478D. Maximal principle
478E. Theorem
478F. Basic examples
478G.
478H. Corollary
478I.
478J. Convolutions and smoothing
478K. Dynkin\'s formula
478L. Theorem
478M. Proposition
478N. Wandering paths
478O. Theorem
478P. Harmonic measures
478Q.
478R. Theorem
478S. Corollary
478T. Corollary
478U.
*478V. Theorem
478X. Basic exercises
478Y. Further exercises
478 Notes and comments
479. Newtonian capacity
479A. Notation
479B. Theorem
479C. Definitions
479D.
479E. Theorem
479F.
479G.
479H.
479I. Proposition
479J.
479K. Lemma
479L.
479M.
479N.
479O. Polar sets
479P.
479Q. Hausdorff measure
479R.
479S.
*479T.
*479U. Theorem
*479V.
*479W.
479X. Basic exercises
479Y. Further exercises
479 Notes and comments
Chapter 48. Gauge integrals
481. Tagged partitions
481A. Tagged partitions and Riemann sums
481B. Notation
481C. Proposition
481D. Remarks
481E. Gauges
481F. Residual sets
481G. Subdivisions
481H. Remarks
481I.
481J. The Henstock integral on a bounded interval (HENSTOCK 63)
481K. The Henstock integral on R
481L. The symmetric Riemann-complete integral (cf. CARRINGTON 72, chap. 3)
481M. The McShane integral on an interval (McSHANE 73)
481N. The McShane integral on a topological space (FREMLIN 95)
481O. Convex partitions in R^r
481P. Box products (cf. MULDOWNEY 87, Prop. 1)
481Q. The approximately continuous Henstock integral (GORDON 94, chap. 16)
481X. Basic exercises
481Y. Further exercises
481 Notes and comments
482. General theory
482A. Lemma
482B. Saks-Henstock Lemma
482C. Definition
482D.
482E. Theorem
482F. Proposition
482G. Proposition
482H. Proposition
482I. Integrating a derivative
482J. Definition
482K. B.Levi\'s theorem
482L. Lemma
482M. Fubini\'s theorem
482X. Basic exercises
482Y. Further exercises
482 Notes and comments
483. The Henstock integral
483A. Definition
483B.
483C. Corollary
483D. Corollary
483E. Definition
483F.
483G. Theorem
483H. Upper and lower derivates
483I. Theorem
483J. Theorem
483K. Proposition
483L. Definition
483M. Proposition
483N. Proposition
483O. Definitions
483P. Elementary results
483Q. Lemma
483R. Theorem
483X. Basic exercises
483Y. Further exercises
483 Notes and comments
484. The Pfeffer integral
484A. Notation
484B. Theorem (TAMANINI & GIACOMELLI 89)
484C. Lemma
484D. Definitions
484E. Lemma
484F. A family of tagged-partition structures
484G. The Pfeffer integral
484H.
484I. Definition
484J.
484K. Lemma
484L. Proposition
484M. Lemma
484N. Pfeffer\'s Divergence Theorem
484O. Differentiating the indefinite integral
484P. Lemma
484Q. Definition
484R. Lemma
484S. Theorem
484X. Basic exercises
484Y. Further exercises
484 Notes and comments
Chapter 49. Further topics
491. Equidistributed sequences
491A. The asymptotic density ideal
491B. Equidistributed sequences
491C.
491D.
491E. Proposition
491F. Theorem
491G. Corollary
491H. Theorem (VEECH 71)
491I. The quotient PN/Z
491J. Lemma
491K. Corollary
491L. Effectively regular measures
491M. Examples
491N. Theorem
491O. Proposition
491P. Proposition
491Q. Corollary
491R.
491S. The asymptotic density filter
491X. Basic exercises
491Y. Further exercises
491Z. Problem
491 Notes and comments
492. Combinatorial concentration of measure
492A. Lemma
492B. Corollary
492C. Lemma
492D. Theorem (TALAGRAND 95)
492E. Corollary
492F.
492G. Lemma (MILMAN & SCHECHTMAN 86)
492H. Theorem (MAUREY 79)
492I. Corollary
492X. Basic exercises
492 Notes and comments
493. Extremely amenable groups
493A. Definition
493B. Proposition
493C. Theorem
493D.
493E. Theorem (PESTOV 02)
493F.
493G. Theorem
493H.
493X. Basic exercises
493Y. Further exercises
493 Notes and comments
494. Groups of measure-preserving automorphisms
494A. Definitions (HALMOS 56)
494B. Proposition
494C. Proposition
494D. Lemma
494E. Theorem (HALMOS 44, ROKHLIN 48)
494F.
494G. Proposition
494H. Proposition
494I.
494J. Lemma
494K. Lemma
494L. Theorem
494M. Lemma
494N. Lemma
494O. Theorem (KITTRELL & TSANKOV 09)
494P. Remark
494Q.
494R.
494X. Basic exercises
494Y. Further exercises
494Z. Problems
494 Notes and comments
495. Poisson point processes
495A. Poisson distributions
495B. Theorem
495C. Lemma
495D. Theorem
495E. Definition
495F. Proposition
495G. Proposition
495H. Lemma
495I. Theorem
495J. Proposition
495K. Proposition
495L.
495M.
495N.
495O. Proposition
495P.
495X. Basic exercises
495Y. Further exercises
495 Notes and comments
496. Maharam submeasures
496A. Definitions
496B. Basic facts
496C. Radon submeasures
496D. Proposition
496E. Theorem
496F. Theorem
496G. Theorem
496H. Theorem
496I. Theorem
496J. Theorem
496K. Proposition
496L. Free products of Maharam algebras
496M. Representing products of Maharam algebras
496X. Basic exercises
496Y. Further exercises
496 Notes and comments
497. Tao\'s proof of Szemerédi\'s theorem
497A. Definitions
497B. Lemma
497C. Lemma
497D. Lemma
497E. Theorem (TAO 07)
497F. Invariant measures on P([I]^{<ω})
497G. Theorem (TAO 07)
497H.
497I. Definition
497J. Theorem (NAGLE RÖDL & SCHACHT 06)
497K. Corollary: the Hypergraph Removal Lemma
497L. Corollary: Szemerédi\'s Theorem (SZEMERÉDI 75)
497M.
497N. Theorem (FURSTENBURG 81)
497X. Basic exercises
497Y. Further exercises
497 Notes and comments
498. Cubes in product spaces
498A. Proposition
498B. Proposition (see BRODSKIĬ 49, EGGLESTON 54)
498C. Proposition (see CIESIELSKI & PAWLIKOWSKI 03)
498X. Basic exercises
498Y. Further exercises
498 Notes and comments
Appendix to Volume 4 - Useful Facts
4A1. Set theory
4A1A. Cardinals again
4A1B. Closed cofinal sets
4A1C. Stationary sets
4A1D. Δ-systems
4A1E. Free sets
4A1F. Selecting subsequences
4A1G. Ramsey\'s theorem
4A1H. The Marriage Lemma again
4A1I. Filters
4A1J. Lemma
4A1K. Theorem
4A1L. Theorem
4A1M. Ostaszewski\'s
4A1N. Lemma
4A1O. The size of σ-algebras
4A1P. An incidental fact
4A2. General topology
4A2A. Definitions
4A2B. Elementary facts about general topological spaces
4A2C. G_δ, F_σ, zero and cozero sets
4A2D. Weight
4A2E. The countable chain condition
4A2F. Separation axioms
4A2G. Compact and locally compact spaces
4A2H. Lindelöf spaces
4A2I. Stone-Čech compactifications
4A2J. Uniform spaces
4A2K. First-countable, sequential and countably tight spaces
4A2L. (Pseudo-)metrizable spaces
4A2M. Complete metric spaces
4A2N. Countable networks
4A2O. Second-countable spaces
4A2P. Separable metrizable spaces
4A2Q. Polish spaces
4A2R. Order topologies
4A2S. Order topologies on ordinals
4A2T. Topologies on spaces of subsets
4A2U. Old friends
4A3. Topological σ-algebras
4A3A. Borel sets
4A3B. (Σ, T)-measurable functions
4A3C. Elementary facts
4A3D. Hereditarily Lindelöf spaces
4A3E. Applications
4A3F. Spaces with countable networks
4A3G. Second-countable spaces
4A3H. Borel sets in Polish spaces
4A3I. Corollary
4A3J. Borel sets in ω_1
4A3K. Baire sets
4A3L. Lemma
4A3M. Product spaces
4A3N. Products of separable metrizable spaces
4A3O. Compact spaces
4A3P. Proposition
4A3Q. Baire property
4A3R. Proposition
*4A3S.
4A3T. Cylindrical σ-algebras
4A3U. Proposition
4A3V. Proposition
4A3W. Càdlàg functions
4A3X. Basic exercises
4A3Y. Further exercises
4A3 Notes and comments
4A4. Locally convex spaces
4A4A. Linear spaces
4A4B. Linear topological spaces
4A4C. Locally convex spaces
4A4D. Hahn-Banach theorem
4A4E. The Hahn-Banach theorem in locally convex spaces
4A4F. The Mackey topology
4A4G. Extreme points
4A4H. Proposition
4A4I. Normed spaces
4A4J. Inner product spaces
4A4K. Hilbert spaces
4A4L. Compact operators
4A4M. Self-adjoint compact operators
4A4N. Max-flow Min-cut Theorem (FORD & FULKERSON 56)
4A5. Topological groups
4A5A. Notation
4A5B. Group actions
4A5C. Examples
4A5D. Definitions
4A5E. Elementary facts
4A5F. Proposition
4A5G. Proposition
4A5H. The uniformities of a topological group
4A5I. Definitions
4A5J. Quotients under group actions, and quotient groups
4A5K. Proposition
4A5L. Theorem
4A5M. Proposition
4A5N. Theorem
4A5O. Proposition
4A5P. Lemma
4A5Q. Metrizable groups
4A5R. Corollary
4A5S. Lemma
*4A5T.
4A6. Banach algebras
4A6A. Definition
4A6B. Stone-Weierstrass Theorem: fourth form
4A6C. Proposition
4A6D. Proposition
4A6E. Proposition
4A6F. Proposition
4A6G. Definition
4A6H. Theorem
4A6I. Theorem
4A6J. Theorem
4A6K. Corollary
4A6L. Exponentiation
4A6M. Lemma
4A6N. Lemma
4A6O. Proposition
4A7. `Later editions only\'
Concordance to chapters 46-49
References for Volume 4
Index to volumes 1-4
Principal topics and results
General index
Measure Theory 5-1_Set-theoretic Measure Theory(2015,329p)D.H.Fremlin_9780953812950
Contents
General introduction
Introduction to Volume 5
Note on second printing
Chapter 51. Cardinal functions
511. Definitions
511A. Pre-ordered sets
511B. Definitions
511C. On the symbol ∞
511D. Definitions
511E. Precalibers
511F. Definitions
511G. Definition
511H. Elementary facts: pre-ordered sets
511I. Elementary facts: Boolean algebras
511J. Elementary facts: ideals of sets
511X. Basic exercises
511Y. Further exercises
511 Notes and comments
512. Galois-Tukey connections
512A. Definitions
512B. Definitions
512C.
512D. Theorem
512E. Examples
512F.
512G. Proposition
512H. Simple products
512I. Sequential compositions
512J. Proposition
512K.
512X. Basic exercises
512 Notes and comments
513. Partially ordered sets
513A.
513B. Theorem
513C. Cofinalities of cardinal functions
513D.
513E. Theorem
513F. Theorem (TUKEY 40)
513G.
513H. Definition
513I. Proposition
*513J. Cofinalities of products
*513K.
*513L. Proposition
*513M. Proposition
*513N. Lemma
*513O. Theorem (SOLECKI & TODORČEVIC 04)
513P.
513X. Basic exercises
513Y. Further exercises
513 Notes and comments
514. Boolean algebras
514A.
514B. Stone spaces
514C.
514D. Theorem
514E. Subalgebras, homomorphic images, products
514F.
514G. Order-preserving functions of Boolean algebras
514H. Regular open algebras
514I. Category algebras
514J.
514K.
514L. The regular open algebra of a pre-ordered set
514M.
514N. Proposition
514O.
514P. Corollary
514Q. Proposition
514R. Corollary
514S. Proposition
514T. Finite-support products
514U. Proposition
514X. Basic exercises
514Y. Further exercises
514. Notes and comments
515. The Balcar-Franĕk theorem
515A. Definition
515B. Lemma
515C. Proposition
515D. Lemma
515E. Lemma (BALCAR & VOITÁŠ 77)
515F. Lemma
515G. Lemma
515H. The Balcar-Franĕk theorem (BALCAR & FRANĔK 82)
515I. Corollary
515J. Corollary
515K.
515L. Theorem (KOPPELBERG 75)
515M. Corollary
515N.
515X. Basic exercises
515Y. Further exercises
515 Notes and comments
516. Precalibers
516A. Definition
516B. Elementary remarks
516C. Theorem
516D. Corollary
516E. Remark
516F.
516G. Corollary
516H. Corollary
516I. Corollary
516J.
516K.
516L. Corollary
516M.
516N. Corollary
516O.
516P. Corollary
516Q.
516R.
516S.
516T.
516U.
516X. Basic exercises
516 Notes and comments
517. Martin numbers
517A. Proposition
517B. Lemma
517C. Lemma
517D. Proposition
517E. Corollary
517F. Proposition
517G. Corollary
517H. Proposition
517I. Proposition
517J. Proposition
517K. Corollary
517L.
517M.
517N. Corollary
517O. Martin cardinals
517P.
517Q. Lemma
517R. Proposition
517S. Proposition
517X. Basic exercises
517Y. Further exercises
517 Notes and comments
518. Freese-Nation numbers
518A. Proposition (FUCHINO KOPPELBERG & SHELAH 96)
518B. Proposition
518C. Corollary
518D. Corollary
518E.
518F. Lemma
518G. Lemma (FUCHINO KOPPELBERG & SHELAH 96)
518H. Lemma
518I. Theorem (FUCHINO & SOUKUP 97)
518J. Lemma
518K. Theorem (FUCHINO GESCHKE SHELAH & SOUKUP 01)
518L.
518M. Theorem
518N. Definition
518O. Lemma
518P. Lemma (GESCHKE 02)
518Q. Corollary
518R. Lemma
518S. Theorem (GESCHKE 02)
518X. Basic exercises
518Y. Further exercises
518 Notes and comments
Chapter 52. Cardinal functions of measure theory
521. Basic theory
521A. Proposition
521B. Proposition
521C.
521D. Proposition
521E.
521F. Proposition
521G. Proposition
521H. Proposition
521I. Corollary
521J.
521K.
521L. Proposition
521M. Proposition
521N. Proposition
521O. Proposition
521P. Proposition
521Q. Free products
521R. Proposition
521S. Proposition
521T.
521X. Basic exercises
521Y. Further exercises
521 Notes and comments
522. Cichoń\'s diagram
522A. Notation
522B. Cichoń\'s diagram
522C. Lemma
522D. Proposition
522E. Proposition
522F. Proposition
522G. Proposition (ROTHBERGER 38A)
522H. Proposition
522I. Proposition
522J. Theorem (see TRUSS 77 and MILLER 81)
522K. Localization
*522L. Lemma
522M. Proposition
522N. Lemma
522O. Proposition
522P. Corollary
522Q. Theorem (BARTOSZYŃSKI 84, RAISONNIER & STERN 85)
522R. The exactness of Cichoń\'s diagram
522S. The cardinal cov M
522T. Martin numbers
*522U. FN(PN)
522V. Cofinalities
522W. Other spaces
522X. Basic exercises
522Y. Further exercises
522 Notes and comments
523. The measure of {0, 1}^{I}
523A. Notation
523B. The basic diagram
523C.
523D.
523E. Additivities
523F. Covering numbers
523G.
523H. Uniformities
523I. Theorem
523J. Corollary (KRASZEWSKI 01)
523K. Corollary (BURKE N05)
523L.
523M. Shrinking numbers
523N. Cofinalities
523O. Cofinalities of the cardinals
523P. The generalized continuum hypothesis
523X. Basic exercises
523Y. Further exercises
523Z. Problem
523 Notes and comments
524. Radon measures
524A. Notation
524B. Proposition
524C. Lemma
524D. Proposition
524E. Proposition
524F. Lemma
524G. Proposition
524H. Corollary
524I. Corollary
524J. Theorem
524K. Corollary
524L.
524M. Theorem
524N. Corollary
524O. Freese-Nation numbers
524P. The Maharam classification
*524Q.
524R.
524S.
524T. Corollary
524X. Basic exercises
524Y. Further exercises
524Z. Problems
524 Notes and comments
525. Precalibers
525A. Notation
525B. Proposition
525C. Theorem
525D. Proposition
525E. Proposition
525F. Proposition
525G.
525H. The structure of B_I
525I. Theorem
525J. Corollary
525K. Proposition
525L.
525M. Proposition
525N. Proposition (ARGYROS & TSARPALIAS 82)
525O.
*525P.
525Q.
525R. Lemma
525S. Theorem (FREMLIN 88)
525T. Corollary (ARGYROS & KALAMIDAS 82)
525X. Basic exercises
525Z. Problem
525 Notes and comments
526. Asymptotic density zero
526A. Proposition
526B. Proposition (FREMLIN 91)
526C.
526D. Lemma
526E. Lemma
526F. Theorem
526G. Corollary
526H.
526I.
526J. Proposition
526K. Proposition
526L. Proposition (MÁTRAI P09)
526M.
526X. Basic exercises
526Y. Further exercises
526 Notes and comments
527. Skew products of ideals
527A. Notation
527B. Skew products of ideals
527C.
527D.
527E. Corollary
527F.
527G. Theorem
527H. Corollary
527I.
527J. Theorem (see FREMLIN 91)
527K. Corollary
527L.
527M.
527N. Lemma
527O. Theorem
527X. Basic exercises
527Y. Further exercises
527 Notes and comments
528. Amoeba algebras
528A. Amoeba algebras
528B.
528C. Proposition
528D. Proposition
528E. Lemma
528F. Proposition
528G. Proposition
528H. Proposition
528I. Definition
528J. Proposition
528K. Theorem (TRUSS 88)
528L.
528M. Lemma
528N. Theorem (BRENDLE 00, 2.3.10; JUDAH & REPICKÝ 95)
528O. Corollary
528P. Proposition
528Q. Proposition
528R. Theorem
528S.
528T. Lemma
528U. Lemma
528V. Theorem
528X. Basic exercises
528Y. Further exercises
528Z. Problems
528 Notes and comments
529. Further partially ordered sets of measure theory
529A. Notation
529B. Proposition
529C. Theorem (FREMLIN 91)
529D. Theorem (FREMLIN 91)
529E. Proposition
529F. Corollary (BRENDLE 00, 2.3.10; BRENDLE 06, Theorem 1)
529G. Reaping numbers (following BRENDLE 00)
529H. Proposition (BRENDLE 00, 2.7; BRENDLE 06, Theorem 5)
529X. Basic exercises
529Y. Further exercises
529 Notes and comments
Chapter 53. Topologies and measures III
531. Maharam types of Radon measures
531A. Proposition
531B.
531C. Lemma
531D. Definition
531E. Proposition
531F. Proposition
531G. Proposition
531H. Remarks
531I. Notation
531J. Lemma
531K. Lemma
531L. Theorem
531M. Proposition (PLEBANEK 97)
531N.
531O.
531P.
531Q.
531R.
531S.
531T. Theorem (FREMLIN 97)
531X. Basic exercises
531Y. Further exercises
531Z. Problems
531 Notes and comments
532. Completion regular measures on {0, 1}^{I}
532A. Definition
532B. Proposition
532C. Remarks
532D. Theorem (FREMLIN & GREKAS 95)
532E. Corollary
532F. Corollary
532G. Proposition
532H. Lemma
532I.
532J. Corollary
532K. Corollary
532L. Corollary
532M.
532N.
532O. Proposition
532P. Proposition
532Q. Proposition
532R.
532S. Proposition
532X. Basic exercises
532Y. Further exercises
532Z. Problems
532 Notes and comments
533. Special topics
533A. Lemma
533B. Corollary
533C. Proposition
533D. Proposition
533E. Corollary
533F. Definition
533G. Lemma
533H. Theorem
533I.
533J. Theorem (see FREMLIN 77)
533X. Basic exercises
533Y. Further exercises
533Z. Problem
533 Notes and comments
534. Hausdorff measures and strong measure zero
534A.
534B. Hausdorff measures
534C. Strong measure zero
534D. Proposition
534E. Rothberger\'s property
534F. Proposition
534G. Corollary
534H. Theorem
534I. Proposition
534J. Proposition (FREMLIN 91)
534K. Corollary
534L. Smz-equivalence
534M. Lemma
534N. Proposition
534O. Large sets with strong measure zero
534P.
534X. Basic exercises
534Y. Further exercises
534Z. Problems
534 Notes and comments
535. Liftings
535A. Notation
535B. Proposition
535C. Proposition
535D.
535E. Proposition
535F.
535G. Corollary (see NEUMANN 31)
535H.
535I. Corollary (see MOKOBODZKI 75)
535J.
535K. Lemma
535L. Lemma
535M. Lemma
535N. Theorem
535O. Linear liftings
535P.
535Q. Proposition
535R. Proposition
535X. Basic exercises
535Y. Further exercises
535Z. Problems
535 Notes and comments
536. Alexandra Bellow\'s problem
536A. The problem
536B. Known cases
536C. Proposition (see TALAGRAND 84, 9-3-3.)
536D. Theorem
536X. Basic exercises
536Y. Further exercises
536 Notes and comments
537. Sierpiński sets, shrinking numbers and strong Fubini theorems
537A. Definitions
537B. Proposition
537C. Entangled sets
537D. Lemma
537E. Lemma
537F. Corollary
537G. Theorem (TODORČEVIĆ 85)
537H. Scalarly measurable functions
537I. Proposition
537J. Corollary
537K.
537L. Corollary
537M.
537N.
537O. Corollary
537P. Corollary
537Q.
537R. Lemma
537S. Proposition
537X. Basic exercises
537Z. Problems
537 Notes and comments
538. Filters and limits
538A. Filters
538B.
538C. Lemma
538D. Finite products of filters
538E.
538F. Ramsey filters
538G. Measure-centering filters
538H. Proposition
538I. Theorem
538J. Proposition
538K.
538L. Theorem
538M. Benedikt\'s theorem (BENEDIKT 98)
538N. Measure-converging filters
538O. The Fatou property
538P. Theorem
538Q. Definition
538R. Proposition
538S. Theorem
538X. Basic exercises
538Y. Further exercises
538Z. Problem
538 Notes and comments
539. Maharam submeasures
539A. The story so far
539B. Proposition
539C. Theorem
539D. Corollary
539E. Proposition (VELIČKOVIĆ 05, BALCAR JECH & PAZÁK 05)
539F. Definition
539G. Proposition
539H. Corollary
539I. Corollary
539J. Theorem
539K.
539L.
539M. Lemma
539N. Theorem (BALCAR JECH & PAZÁK 05, VELIČKOVIĆ 05)
539O. Corollary
539P.
539Q. Reflection principles
539R. Exhaustivity rank
539S. Elementary facts
539T. The rank of a Maharam algebra
539U. Theorem
539X. Basic exercises
539Y. Further exercises
539Z. Problems
539 Notes and comments
Concordance to chapters 51-53
Measure Theory 5-2_Set-theoretic Measure Theory(2015,411p)D.H.Fremlin_9780953812967
Contents
Chapter 54. Real-valued-measurable cardinals
541. Saturated ideals
541A. Definition
541B. Proposition
541C. Proposition
541D. Lemma
541E. Corollary
541F. Lemma
541G. Definition
541H. Proposition
541I. Lemma
541J. Theorem (SOLOVAY 71)
541K. Lemma
541L. Theorem
541M. Definition
541N. Theorem
541O. Lemma
541P. Theorem (TARSKI 45, SOLOVAY 71)
541Q. Theorem
541R. Corollary
541S. Lemma
541X. Basic exercises
541Y. Further exercises
541 Notes and comments
542. Quasi-measurable cardinals
542A. Definition
542B. Proposition
542C. Proposition
542D. Proposition
542E. Theorem (GITIK & SHELAH 93)
542F. Corollary
542G. Corollary
542H. Lemma
542I. Theorem (SHELAH 96)
542J. Corollary
542X. Basic exercises
542Y. Further exercises
542 Notes and comments
543. The Gitik-Shelah theorem
543A. Definitions
543B.
543C. Theorem (see KUNEN N70)
543D. Corollary
543E. The Gitik-Shelah theorem (GITIK & SHELAH 89, GITIK & SHELAH 93)
543F. Theorem
543G. Corollary
543H. Corollary
543I. Corollary
543J. Proposition
543K. Proposition
543L. Proposition
543X. Basic exercises
543Y. Further exercises
543Z. Problems
543 Notes and comments
544. Measure theory with an atomlessly-measurable cardinal
544A. Notation
544B. Proposition
544C. Theorem (KUNEN N70)
544D. Corollary
544E. Theorem (KUNEN N70)
544F. Theorem (KUNEN N70)
544G. Proposition
544H. Corollary
544I.
544J. Proposition (ZAKRZEWSKI 92)
544K. Proposition
544L. Corollary
544M. Theorem
544N. Cichoń\'s diagram and other cardinals
544X. Basic exercises
544Y. Further exercises
544Z. Problems
544 Notes and comments
545. PMEA and NMA
545A. Theorem
545B. Definition
545C. Proposition
545D. Definition
545E. Proposition
545F. Proposition
545G. Corollary
545X. Basic exercises
545Y. Further exercises
545 Notes and comments
546 Power set σ-quotient algebras
546A.
546B. Lemma
546C.
546D.
546E.
546F. Corollary
546G. The Gitik-Shelah theorem for Cohen algebras
546H.
546I. Corollary
546J.
546K. Lemma
546L.
546M. Theorem
546N. Lemma
546O. Lemma
546P. Theorem
546Q. Corollary
546X. Basic exercises
546Y. Further exercises
546Z. Problems
546 Notes and comments
547. Disjoint refinements of sequences of sets
547A. Lemma
547B. Lemma
547C. Lemma
547D. Lemma
547E. Lemma
547F. Theorem
547G. Corollary
547H.
547I. Proposition
547J. Corollary
547X. Basic exercises
547Z. Problems
547 Notes and comments
Chapter 55. Possible worlds
551. Forcing with quotient algebras
551A. Definition
551B. Definition
551C. Definition
551D. Definition
551E. Proposition
551F. Proposition
551G.
551H. Examples
551I. Theorem
551J. Corollary
551K.
551L. Remark
551M.
551N. Proposition
551O. Measure algebras
551P. Theorem
551Q. Iterated forcing
551R. Extending filters
551X. Basic exercises
551Y. Further exercises
551 Notes and comments
552. Random reals I
552A. Notation
552B. Theorem
552C. Theorem
552D. Lemma
552E. Theorem
552F. Theorem
552G. Theorem
552H. Theorem
552I. Theorem
552J. Theorem
552K. Lemma
552L. Lemma
552M. Proposition
552N. Theorem (CARLSON 84)
552O. Proposition
552P. Theorem
552X. Basic exercises
552Y. Further exercises
552 Notes and comments
553. Random reals II
553A. Notation
553B. Lemma
553C. Proposition
553D. Remark
553E. Proposition
553F. Corollary
553G. Lemma
553H. Theorem
553I. Lemma
553J. Theorem
553K.
553L. Lemma
553M. Proposition (LAVER 87)
553N. Proposition
553O.
553X. Basic exercises
553Y. Further exercises
553Z. Problem
553 Notes and comments
554. Cohen reals
554A. Notation
554B. Theorem
554C. Definition
554D. Proposition
554E. Theorem
554F. Corollary
554G. Theorem
554H. Corollary
554I. Theorem (CARLSON FRANKIEWICZ & ZBIERSKI 94)
554X. Basic exercises
554Y. Further exercises
554 Notes and comments
555. Solovay\'s construction of real-valued-measurable cardinals
555A. Notation
555B. Theorem
555C. Theorem
555D. Corollary (SOLOVAY 71)
555E. Theorem
555F. Proposition
555G. Cohen forcing
555H. Corollary
555I.
555J. Lemma
555K. Główczyński\'s example (GŁÓWCZYŃSKI 91, BALCAR JECH & PAZÁK 05, GŁÓWCZYŃSKI 08)
555L. Supercompact cardinals and the normal measure axiom
555M. Proposition
555N. Theorem (PRIKRY 75, FLEISSNER 91)
555O.
555X. Basic exercises
555Y. Further exercises
555Z. Problems
555 Notes and comments
556. Forcing with Boolean subalgebras
556A. Forcing with Boolean subalgebras
556B. Theorem
556C. Theorem
556D. Regularly embedded subalgebras
556E. Proposition
556F. Quotient forcing
556G. Proposition
556H. L^0(\\mathfrak{A})
556I. Proposition
556J. Theorem
556K. Theorem
556L. Relatively independent subalgebras
556M. Laws of large numbers
556N. Dye\'s theorem
556O.
556P. Kawada\'s theorem
556Q.
556R. Proposition
556S. Theorem (FARAH 06)
556X. Basic exercises
556Y. Further exercises
556 Notes and comments
Chapter 56. Choice and determinacy
561. Analysis without choice
561A. Set theory without choice
561B. Real analysis without choice
561C.
561D. Tychonoff\'s theorem
561E. Baire\'s theorem
561F. Stone\'s Theorem
561G. Haar measure
561H. Kakutani\'s theorem
561I. Hilbert spaces
561X. Basic exercises
561Y. Further exercises
561 Notes and comments
562. Borel codes
562A. Trees
562B. Coding sets with trees
562C.
562D. Proposition
562E. Proposition
*562F.
562G. Resolvable sets
562H. Proposition
562I. Theorem
562J. Codable families of sets
562K. Proposition
562L. Codable Borel functions
562M. Theorem
562N. Proposition
562O. Remarks
562P. Codable Borel equivalence
562Q. Resolvable functions
562R. Theorem
562S. Codable families of codable functions
562T. Codable Baire sets
562U. Proposition
562V.
562X. Basic exercises
562Y. Further exercises
562 Notes and comments
563. Borel measures without choice
563A. Definitions
563B. Proposition
563C. Corollary
563D.
563E. Lemma
563F. Proposition
563G. Proposition
563H. Theorem
563I. Theorem
563J. Baire-coded measures
563K. Proposition
563L. Proposition
563M. Measure algebras
563N. Theorem
563X. Basic exercises
563Z. Problem
563 Notes and comments
564. Integration without choice
564A. Definitions
564B. Lemma
564C. Definition
564D. Lemma
564E. Theorem
564F.
564G. Integration over subsets
564H. Theorem
564I. Riesz Representation Theorem
564J. The space L^1
564K.
564L. Radon-Nikodým theorem
564M. Inverse-measure-preserving functions
564N. Product measures
564O. Theorem
564X. Basic exercises
564Y. Further exercises
564 Notes and comments
565. Lebesgue measure without choice
565A. Definitions
565B. Proposition
565C. Lemma
565D. Definition
565E. Proposition
565F. Vitali\'s Theorem
565G. Proposition
565H. Corollary
565I. Lemma
565J. Lemma
565K. Theorem
565L. Lemma
565M. Theorem
565N. Hausdorff measures
565O. Theorem
565X. Basic exercises
565Y. Further exercises
565 Notes and comments
566. Countable choice
566A.
566B. Volume 1
566C. Volume 2
566D. Exhaustion
566E.
566F. Atomless algebras
566G. Vitali\'s theorem
566H. Bounded additive functionals
566I. Infinite products
566J.
566K. Volume 3
566L. The Loomis-Sikorski theorem
566M. Measure algebras
566N. Characterizing the usual measure on {0, 1}^N
566O. Boolean values
566P. Weak compactness
566Q. Theorem [AC(ω)]
566R. Automorphisms of measurable algebras
566S. Volume 4
566T.
566U. Dependent choice
566X. Basic exercises
566Y. Further exercises
566Z. Problem
566 Notes and comments
567. Determinacy
567A. Infinite games
567B. Theorem
567C. The axiom of determinacy
567D. Theorem (MYCIELSKI 64)
567E. Consequences of AC(R; ω)
567F. Lemma (see MYCIELSKI & ŚWIERCZKOWSKI 64) [AC(R; ω)]
567G. Theorem [AD]
567H. Theorem
567I. Proposition [AC(R; ω)]
567J. Proposition [AD]
567K. Theorem [AD+AC(ω)]
567L. Theorem (R.M.Solovay) [AD]
567M. Theorem (MOSCHOVAKIS 70) [AD]
567N. Theorem (MARTIN 70) [AC]
567O. Corollary [AC]
567X. Basic exercises
567Y. Further exercises
567 Notes and comments
Appendix to Volume 5 - Useful Facts
5A1. Set theory
5A1A. Order types
5A1B. Ordinal arithmetic
5A1C. Well-founded sets
5A1D. Trees
5A1E. Cardinal arithmetic
5A1F. Three fairly simple facts
5A1G. Partition calculus - The Erdös-Rado theorem
5A1H. Δ-systems and free sets
5A1I. Remarks
5A1J. Lemma
5A1K. Lemma
5A1L. Definitions
5A1M. Lemma
5A1N. Almost-square-sequences
5A1O. Corollary
5A2. Pcf theory
5A2A. Reduced products
5A2B. Theorem
5A2C. Theorem
5A2D. Definitions
5A2E. Lemma
5A2F. Lemma
5A2G. Theorem
5A2H. Lemma
5A2I. Lemma
5A3. Forcing
5A3A. Forcing notions
5A3B. Forcing languages
5A3C. The Forcing Relation (KUNEN 80, VII.3.3)
5A3D. The Forcing Theorem
5A3E. More notation
5A3F. Boolean truth values
5A3G. Concerning š
5A3H. Names for functions
5A3I. Regular open algebras
5A3J.
5A3K. Lemma
5A3L. Real numbers in forcing languages
5A3M. Forcing with Boolean algebras
5A3N. Ordinals and cardinals
5A3O. Iterated forcing (KUNEN 80, VIII.5.2)
5A3P. Martin\'s axiom
5A3Q. Countably closed forcings
5A3 Notes and comments
5A4. General topology
5A4A. Definitions
5A4B. Proposition
5A4C. Compactness
5A4D. Vietoris topologies
5A4E. Category and the Baire property
5A4F. Normal and paracompact spaces
5A4G. Baire σ-algebras
5A4H. Proposition
5A4I. Old friends
5A5. Real analysis
5A5A. Entire functions
5A6. Special axioms
5A6A. The generalized continuum hypothesis
5A6B. L, 0^# and Jensen\'s Covering Lemma
5A6C. Theorem
5A6D. Square principles
5A6E. Lemma
5A6F. Chang\'s transfer principle
5A6G. Todorčević\'s p-ideal dichotomy
*5A6H. Analytic P-ideals
5A6I. u, g and the filter dichotomy
*5A6J. Proposition (BLASS & LAFLAMME 89)
References for Volume 5
Index to volumes 1-5
Principal topics and results
General index
A
B
C
D
E
F
G
H
I
J・K
L
M
N
O
P
Q・R
S
T
U
V・W
Z
Subject Index
A
B・C
D・E・F
G・H・I・J・K・L
M
N・O・P・Q・R
S・T・U・V・W・Z
α・β・γ・δ・θ・λ・μ・ν・π・σ
τ・υ・φ・χ・ψ・ω
math symbols