0$ or (iii) the $X_n$ are identically distributed.} \section{274}{The Central Limit Theorem}{13.4.10}{360}{} {Normally distributed r.vs; Lindeberg's condition for the Central Limit Theorem; corollaries; estimating $\int_{\alpha}^{\infty}e^{-x^2/2}dx$.} \section{275}{Martingales}{3.12.12}{371}{} {Sequences of $\sigma$-algebras, and martingales adapted to them; up-crossings; Doob's Martingale Convergence Theorem; uniform integrability, $\|\,\|_1$-\vthsp{}convergence and martingales as sequences of conditional expectations; reverse martingales; stopping times.} \section{276}{Martingale difference sequences}{16.4.13}{382}{} {Martingale difference sequences; strong law of large numbers for m.d.ss.; Koml\'os' theorem.} \wheader{}{10}{4}{4}{100pt} Chapter 28: Fourier analysis \chapintrosection{17.1.15}{390}{} \section{281}{The Stone-Weierstrass theorem}{4.12.12}{390}{} {Approximating a function on a compact set by members of a given lattice or algebra of functions; real and complex cases; approximation by polynomials and trigonometric functions; Weyl's Equidistribution Theorem in $[0,1]^r$.} \section{282}{Fourier series}{24.9.09}{401}{} {Fourier and Fej\'er sums; Dirichlet and Fej\'er kernels; Riemann-Lebesgue lemma; uniform convergence of Fej\'er sums of a continuous function; a.e.\ and $\|\,\|_1$-convergence of Fej\'er sums of an integrable function; $\|\,\|_2$-convergence of Fourier sums of a square-integrable function; convergence of Fourier sums of a differentiable or b.v.\ function; convolutions and Fourier coefficients.} \section{283}{Fourier transforms I}{31.3.13}{419}{} {Fourier and inverse Fourier transforms; elementary properties; $\int_0^{\infty}\bover1x\sin{x}\,dx \ifdim\pagewidth>467pt\penalty-100\fi =\bover12\pi$; the formula $\varhatf\varcheck{\phantom{h}}=f$ for differentiable and b.v.\ $f$; convolutions; $e^{-x^2/2}$; ${\int}f\times\varhat{g}=\int\varhatf{\times}g$.} \section{284}{Fourier transforms II}{30.8.13}{434}{} {Test functions; $\varhat{h}\varcheck{\phantom{h}}=h$; tempered functions; tempered functions which represent each other's transforms; convolutions; square-integrable functions; Dirac's delta function.} \section{285}{Characteristic functions}{18.9.14}{450}{} {The characteristic function of a distribution; independent r.vs; the normal distribution; the vague topology on the space of distributions, and sequential convergence of characteristic functions; Poisson's theorem; convolutions of distributions.} \section{*286}{Carleson's theorem}{23.4.10}{464}{} {The Hardy-Littlewood Maximal Theorem; the Lacey-Thiele proof of Carleson's theorem for square-integrable functions on $\Bbb{R}$ and $\ocint{-\pi,\pi}$.} \wheader{}{10}{4}{4}{100pt} Appendix to Volume 2 \chapintrosection{3.8.15}{494}{} \section{2A1}{Set theory}{20.1.13}{494}{} {Ordered sets; transfinite recursion; ordinals; initial ordinals; Schr\"oder-\vthsp{}Bernstein theorem; filters; Axiom of Choice; Zermelo's Well-Ordering Theorem; Zorn's Lemma; ultrafilters; a theorem in combinatorics.} \section{2A2}{The topology of Euclidean space}{30.11.09}{500}{} {Closures; continuous functions; compact sets; open sets in $\Bbb{R}$.} \section{2A3}{General topology}{25.7.07}{503}{} {Topologies; continuous functions; subspace topologies; closures and interiors; Hausdorff topologies; pseudometrics; convergence of sequences; compact spaces; cluster points of sequences; convergence of filters; lim sup and lim inf; product topologies; dense subsets.} \section{2A4}{Normed spaces}{4.3.14}{511}{} {Normed spaces; linear subspaces; Banach spaces; bounded linear operators; dual spaces; extending a linear operator from a dense subspace; normed algebras.} \section{2A5}{Linear topological spaces}{13.11.07}{514}{} {Linear topological spaces; topologies defined by functionals; convex sets; completeness; weak topologies.} \section{2A6}{Factorization of matrices}{10.11.14}{517}{} {Determinants; orthonormal families; $T=PDQ$ where $D$ is diagonal and $P$, $Q$ are orthogonal.} \wheader{}{10}{4}{4}{100pt} Concordance \pagereference{519}{} \medskip References for Volume 2 \vtmpb{17.12.12}\pagereference{521}{} \medskip Index to Volumes 1 and 2 \qquad Principal topics and results \pagereference{523}{} \qquad General index \pagereference{527}{} \bigskip \noindent Volume 3: Measure Algebras \medskip \noindent Part I \medskip \wheader{}{10}{4}{4}{100pt} Introduction to Volume 3 \vtmpb{10.9.13}\pagereference{11}{} %wheader parameters: #1 new paragraph name % #2#3#4 \vskip parameters if page break not forced (no. of points) % #5 spare height demanded if page break not to be forced \wheader{}{10}{4}{4}{100pt} Chapter 31: Boolean algebras \chapintrosection{29.10.12}{13}{} \section{311}{Boolean algebras}{15.10.08}{13}{} {Boolean rings and algebras; ideals and ring homomorphisms to $\Bbb{Z}_2$; Stone's theorem; the operations $\Bcup$, $\Bcap$, $\Bsymmdiff$, $\Bsetminus$ and the relation $\Bsubseteq$; partitions of unity; topology of the Stone space; Boolean algebras as complemented distributive lattices.} \section{312}{Homomorphisms}{29.5.07}{21}{} {Subalgebras; ideals; Boolean homomorphisms; the ordering determines the ring structure; quotient algebras; extension of homomorphisms; homomorphisms and Stone spaces.} \section{313}{Order-continuity}{8.6.11}{29}{} {General distributive laws; order-closed sets; order-closures; Monotone Class Theorem; order-preserving functions; order-continuity; order-dense sets; order-continuous Boolean homomorphisms; and Stone spaces; regularly embedded subalgebras; upper envelopes.} \section{314}{Order-completeness}{26.7.07}{39}{} {Dedekind completeness and $\sigma$-completeness; quotients, subalgebras, principal ideals; order-continuous homomorphisms; extension of homomorphisms; Loomis-Sikorski representation of a $\sigma$-complete algebra as a quotient of a $\sigma$-algebra of sets; regular open algebras; Stone spaces; Dedekind completion of a Boolean algebra.} \section{315}{Products and free products}{13.11.12}{49}{} {Simple product of Boolean algebras; free product of Boolean algebras; algebras of sets and their quotients; projective and inductive limits.} \section{316}{Further topics}{26.1.09}{59}{} {The countable chain condition; weak $(\sigma,\infty)$-distributivity; Stone spaces; atomic and atomless Boolean algebras; homogeneous Boolean algebras.} \wheader{}{10}{4}{4}{100pt} Chapter 32: Measure algebras \chapintrosection{21.7.11}{68}{} \section{321}{Measure algebras}{3.1.11}{68}{} {Measure algebras; elementary properties; the measure algebra of a measure space; Stone spaces.} \section{322}{Taxonomy of measure algebras}{24.4.06}{71}{} {Totally finite, $\sigma$-finite, semi-finite and localizable measure algebras; relation to corresponding types of measure space; completions and c.l.d.\ versions of measures; semi-finite measure algebras are \wsid; subspace measures and indefinite-integral measures; simple products of measure algebras; Stone spaces of localizable measure algebras; localizations of semi-finite measure algebras.} \section{323}{The topology of a measure algebra}{20.7.06}{81}{} {Defining a topology and uniformity on a measure algebra; continuity of algebraic operations; order-closed sets; Hausdorff and metrizable topologies, complete uniformities; closed subalgebras; products.} \section{324}{Homomorphisms}{6.2.02}{87}{} {Homomorphisms induced by measurable functions; order-continuous and continuous homomorphisms; the topology of a semi-finite measure algebra is determined by the algebraic structure; measure-preserving homomorphisms.} \section{325}{Free products and product measures}{30.8.06}{93}{} {The measure algebra of a product measure; the localizable measure algebra free product of two semi-finite measure algebras; the measure algebra of a product of probability measures; the probability algebra free product of probability algebras; factorizing through subproducts.} \section{326}{Additive functionals on Boolean algebras}{21.5.11}{102}{} {Additive, countably additive and completely additive functionals; Jordan decomposition; Hahn decomposition; Liapounoff's convexity theorem; the region $\Bvalue{\mu>\nu}$.} \section{327}{Additive functionals on measure algebras}{13.7.11}{114}{} {Absolutely continuous and continuous additive functionals; Radon-Nikod\'ym theorem; the standard extension of a continuous additive functional on a closed subalgebra.} \section{*328}{Reduced products and other constructions}{2.6.09}{120}{} {Reduced products of probability algebras; inductive and projective limits; converting homomorphisms into automorphisms.} \wheader{}{10}{4}{4}{100pt} Chapter 33: Maharam's theorem \chapintrosection{17.11.10}{127}{} \section{331}{Maharam types and homogeneous measure algebras}{1.2.05}{127}{} {Relatively atomless algebras; one-step extension of measure-preserving homomorphisms; Maharam type of a measure algebra; \Mth\ probability algebras of the same Maharam type are isomorphic; the measure algebra of $\{0,1\}^{\kappa}$ is homogeneous.} \section{332}{Classification of localizable measure algebras}{19.3.05}{135}{} {Any localizable measure algebra is isomorphic to a simple product of homogeneous totally finite algebras; complete description of isomorphism types; closed subalgebras.} \section{333}{Closed subalgebras}{27.6.08}{145}{} {Relative Maharam types; extension of measure-preserving Boolean homomorphisms; complete classification of closed subalgebras of probability algebras as triples $(\frak{A},\bar\mu,\frak{C})$; fixed-point subalgebras.} \section{334}{Products}{26.9.08}{160}{} {Maharam types of product measures; infinite powers of probability spaces are Maharam-type-homogeneous.} \wheader{}{10}{4}{4}{100pt} Chapter 34: Liftings \chapintrosection{28.5.07}{163}{} \section{341}{The lifting theorem}{9.4.10}{163}{} {Liftings and lower densities; strictly localizable spaces have lower densities; construction of a lifting from a density; complete strictly localizable spaces have liftings; liftings and Stone spaces.} \section{342}{Compact measure spaces}{9.7.10}{174}{} {Inner regular measures; compact classes; compact and locally compact measures; perfect measures.} \section{343}{Realization of homomorphisms}{17.11.10}{181}{} {Representing homomorphisms between measure algebras by functions; possible when target measure space is locally compact; countably separated measures and uniqueness of representing functions; the split interval; perfect measures.} \section{344}{Realization of automorphisms}{22.3.06}{189}{} {Simultaneously representing groups of automorphisms of measure algebras by functions -- Stone spaces, countably separated measure spaces, measures on $\{0,1\}^I$; characterization of Lebesgue measure as a measure space; strong homogeneity of usual measure on $\{0,1\}^I$.} \section{345}{Translation-invariant liftings}{27.6.06}{198}{} {Translation-invariant liftings on $\Bbb{R}^r$ and $\{0,1\}^I$; there is no t.-i.\ Borel lifting on $\Bbb{R}$.} \section{346}{Consistent liftings}{17.12.10}{206}{} {Liftings of product measures which respect the product structure; translation-invariant liftings on $\{0,1\}^I$; products of \Mth\ probability spaces; lower densities respecting product structures; consistent liftings; the Stone space of Lebesgue measure.} \wheader{}{10}{4}{4}{0pt} Concordance to part I \pagereference{216}{} %213 pp \noindent Part II \wheader{}{10}{4}{4}{100pt} Chapter 35: Riesz spaces \chapintrosection{4.9.09}{8}{} \section{351}{Partially ordered linear spaces}{16.10.07}{8}{} {Partially ordered linear spaces; positive cones; suprema and infima; positive linear operators; order-continuous linear operators; Riesz homomorphisms; quotient spaces; reduced powers; representation of p.o.l.ss as subspaces of reduced powers of $\Bbb{R}$; Archimedean spaces.} \section{352}{Riesz spaces}{22.9.03}{14}{} {Riesz spaces; identities; general distributive laws; Riesz homomorphisms; Riesz subspaces; order-dense subspaces and order-continuous operators; bands; the algebra of complemented bands; the algebra of projection bands; principal bands; $f$-algebras.} \section{353}{Archimedean and Dedekind complete Riesz spaces}{26.1.02}{25}{} {Order-dense subspaces; bands; Dedekind ($\sigma$)-complete spaces; order-closed Riesz subspaces; order units; $f$-algebras.} \section{354}{Banach lattices}{18.8.08}{32}{} {Riesz norms; Fatou norms; the Levi property; order-continuous norms; order-unit norms; $M$-spaces; are isomorphic to $C(X)$ for compact Hausdorff $X$; $L$-spaces; uniform integrability in $L$-spaces.} \section{355}{Spaces of linear operators}{1.12.07}{41}{}{Order-bounded linear operators; the space $\eurm{L}^{\sim}(U;V)$; order-continuous operators; extension of order-continuous operators; the space $\eurm{L}^{\times}(U;V)$; order-continuous norms.} \section{356}{Dual spaces}{26.1.02}{49}{} {The spaces $U^{\sim}$, $U^{\times}$, $U^*$; biduals, embeddings $U{\to}V^{\times}$ where $V{\subseteq}U^{\sim}$; perfect Riesz spaces; $L$- and $M$-spaces; uniformly integrable sets in the dual of an $M$-space; relative weak compactness in $L$-spaces.} \wheader{}{10}{4}{4}{100pt} Chapter 36: Function spaces \chapintrosection{2.2.02}{59}{} \section{361}{$S$}{6.2.08}{60}{} {Additive functions on Boolean rings; the space $S(\frak{A})$; universal mapping theorems for linear operators on $S$; the map $T_{\pi}:S(\frak{A}){\to}S(\frak{B})$ induced by a ring homomorphism $\pi:\frak{A}\to\frak{B}$; projection bands in $S(\frak{A})$; identifying $S(\frak{A})$ when $\frak{A}$ is a quotient of an algebra of sets.} \section{362}{$S^{\sim}$}{31.12.10}{71}{} {Bounded additive functionals on $\frak{A}$ identified with order-bounded linear functionals on $S(\frak{A})$; the $L$-space $S^{\sim}$ and its bands; countably additive, completely additive, absolutely continuous and continuous functionals; uniform integrability in $S^{\sim}$.} \section{363}{$L^{\infty}$}{4.3.08}{81}{} {The space $L^{\infty}(\frak{A})$, as an $M$-space and $f$-algebra; universal mapping theorems for linear operators on $L^{\infty}$; $T_{\pi}:L^{\infty}(\frak{A}){\to}L^{\infty}(\frak{B})$; representing $L^{\infty}$ when $\frak{A}$ is a quotient of an algebra of sets; integrals with respect to finitely additive functionals; projection bands in $L^{\infty}$; $(L^{\infty})^{\sim}$ and its bands; Dedekind completeness of $\frak{A}$ and $L^{\infty}$; representing $\sigma$-complete $M$-spaces; the generalized Hahn-Banach theorem; the Banach-Ulam problem.} \section{364}{$L^0$}{16.7.11}{95}{} {The space $L^0(\frak{A})$; representing $L^0$ when $\frak{A}$ is a quotient of a $\sigma$-algebra of sets; algebraic operations on $L^0$; action of Borel measurable functions on $L^0$; identifying $L^0(\frak{A})$ with $L^0(\mu)$ when $\frak{A}$ is a measure algebra; embedding $S$ and $L^{\infty}$ in $L^0$; suprema and infima in $L^0$; Dedekind completeness in $\frak{A}$ and $L^0$; multiplication in $L^0$; projection bands; $T_{\pi}:L^0(\frak{A}){\to}L^0(\frak{B})$; when $\pi$ represented by a $(\Tau,\Sigma)$-measurable function; simple products; *regular open algebras; *the space $C^{\infty}(X)$.} \section{365}{$L^1$}{20.1.15}{115}{} {The space $L^1(\frak{A},\bar\mu)$; identification with $L^1(\mu)$; $\int_au$; the Radon-Nikod\'ym theorem again; ${\int}w{\times}T_{\pi}u\,d\bar\nu={\int}u\,d\bar\mu$; additive functions on $\frak{A}$ and linear operators on $L^1$; the duality between $L^1$ and $L^{\infty}$; $T_{\pi}:L^1(\frak{A},\bar\mu){\to}L^1(\frak{B},\bar\nu)$ and $P_{\pi}:L^1(\frak{B},\bar\nu){\to}L^1(\frak{A},\bar\mu)$; conditional expectations; bands in $L^1$; varying $\bar\mu$.} \section{366}{$L^p$}{10.11.08}{131}{} {The spaces $L^p(\frak{A},\bar\mu)$; identification with $L^p(\mu)$; $L^q$ as the dual of $L^p$; the spaces $M^0$ and $M^{1,0}$; $T_{\pi}:M^0(\frak{A},\bar\mu){\to}M^0(\frak{B},\bar\nu)$ and $P_{\pi}:M^{1,0}(\frak{B},\bar\nu){\to}M^{1,0}(\frak{A},\bar\mu)$; conditional expectations; the case $p=2$; spaces $L^p_{\Bbb{C}}(\frak{A},\bar\mu)$.} \section{367}{Convergence in measure}{18.9.03}{141}{} {Order*-convergence of sequences in lattices; in Riesz spaces; in Banach lattices; in quotients of spaces of measurable functions; in $C(X)$; Lebesgue's Dominated Convergence Theorem and Doob's Martingale Theorem; convergence in measure in $L^0(\frak{A})$; and pointwise convergence; defined by the Riesz space structure; positive linear operators on $L^0$; convergence in measure and the canonical projection $(L^1)^{**}{\to}L^1$; the set of independent families of random variables.} \section{368}{Embedding Riesz spaces in $L^0$}{16.9.09}{157}{} {Extension of order-continuous Riesz homomorphisms into $L^0$; representation of Archimedean Riesz spaces as subspaces of $L^0$; Dedekind completion of Riesz spaces; characterizing $L^0$ spaces as Riesz spaces; \wsid\ Riesz spaces.} \section{369}{Banach function spaces}{5.11.03}{167}{} {Riesz spaces separated by their order-continuous duals; representing $U^{\times}$ when $U{\subseteq}L^0$; Kakutani's representation of $L$-spaces as $L^1$ spaces; extended Fatou norms; associate norms; $L^{\tau'}\cong(L^{\tau})^{\times}$; Fatou norms and convergence in measure; $M^{\infty,1}$ and $M^{1,\infty}$, $\|\,\|_{\infty,1}$ and $\|\,\|_{1,\infty}$; $L^{\tau_1}+L^{\tau_2}$.} \wheader{}{10}{4}{4}{100pt} Chapter 37: Linear operators between function spaces \chapintrosection{6.11.03}{182}{} \section{371}{The Chacon-Krengel theorem}{13.12.06}{182}{} {$\eurm{L}^{\sim}(U;V)=\eurm{L}^{\times}(U;V)=\eurm{B}(U;V)$ for $L$-spaces $U$ and $V$; the class $\Cal{T}^{(0)}_{\bar\mu,\bar\nu}$ of $\|\,\|_1$-decreasing, $\|\,\|_{\infty}$-decreasing linear operators from $M^{1,0}(\frak{A},\bar\mu)$ to $M^{1,0}(\frak{B},\bar\nu)$.} \section{372}{The ergodic theorem}{7.12.08}{186}{} {The Maximal Ergodic Theorem and the Ergodic Theorem for operators in $\Cal{T}^{(0)}_{\bar\mu,\bar\mu}$; for \imp\ functions $\phi:X{\to}X$; limit operators as conditional expectations; applications to continued fractions; mixing and ergodic transformations.} \section{373}{Decreasing rearrangements}{10.2.02}{202}{}{The classes $\Cal{T}$, $\Cal{T}^{\times}$; the space $M^{0,\infty}$; decreasing rearrangements $u^*$; $\|u^*\|_p=\|u\|_p$; $\int|Tu{\times}v|\le{\int}u^*{\times}v^*$ if $T\in\Cal{T}$; the very weak operator topology and compactness of $\Cal{T}$; $v$ is expressible as $Tu$, where $T\in\Cal{T}$, iff $\int_0^tv^*\le\int_0^tu^*$ for every $t$; finding $T$ such that ${\int}Tu{\times}v={\int}u^*{\times}v^*$; the adjoint operator from $\Cal{T}^{(0)}_{\bar\mu,\bar\nu}$ to $\Cal{T}^{(0)}_{\bar\nu,\bar\mu}$.} \section{374}{Rearrangement-invariant spaces}{15.6.09}{219}{} {$\Cal{T}$-invariant subspaces of $M^{1,\infty}$, and $\Cal{T}$-invariant extended Fatou norms; relating $\Cal{T}$-invariant norms on different spaces; rearrangement-invariant sets and norms; when rearrangement-invariance implies $\Cal{T}$-invariance.} \section{375}{Kwapien's theorem}{30.1.10}{229}{} {Linear operators on $L^0$ spaces; if $\frak{B}$ is measurable, a positive linear operator from $L^0(\frak{A})$ to $L^0(\frak{B})$ can be assembled from Riesz homomorphisms.} \section{376}{Integral operators}{8.4.10}{235}{} {Kernel operators; free products of measure algebras and tensor products of $L^0$ spaces; tensor products of $L^1$ spaces; abstract integral operators (i) as a band in $\eurm{L}^{\times}(U;V)$ (ii) represented by kernels belonging to $L^0(\frak{A}\tensorhat\frak{B})$ (iii) as operators converting weakly convergent sequences into order*-convergent sequences; operators into $M$-spaces or out of $L$-spaces.} \section{377}{Function spaces of reduced products}{30.12.09}{251}{} {Measure-bounded Boolean homomorphisms on products of probability algebras; associated maps on subspaces of $\prod_{i\in{I}}L^0(\frak{A}_i)$ and $\prod_{i\in{I}}L^p(\frak{A}_i)$; reduced powers; universal mapping theorems for function spaces on projective and inductive limits of probability algebras.} \wheader{}{10}{4}{4}{100pt} Chapter 38: Automorphisms \chapintrosection{15.8.08}{264}{} \section{381}{Automorphisms of Boolean algebras}{19.7.06}{264}{} {Assembling an automorphism; elements supporting an automorphism; periodic and aperiodic parts; full and countably full subgroups; recurrence; induced automorphisms of principal ideals; Stone spaces; cyclic automorphisms.} \section{382}{Factorization of automorphisms}{15.8.06}{276}{} {Separators and transversals; Frol\'{\i}k's theorem; exchanging involutions; expressing an automorphism as the product of three involutions; subgroups of $\Aut\frak{A}$ with many involutions; normal subgroups of full groups with many involutions; simple automorphism groups.} \section{383}{Automorphism groups of measure algebras}{9.11.14}{290}{} {Measure-preserving automorphisms as products of involutions; normal subgroups of $\Aut\frak{A}$ and $\Aut_{\bar\mu}\frak{A}$; conjugacy in $\Aut\frak{A}$ and $\Aut_{\bar\mu}\frak{A}$.} \section{384}{Outer automorphisms}{5.11.14}{294}{} {If $G\le\Aut\frak{A}$, $H\le\Aut\frak{B}$ have many involutions, any isomorphism between $G$ and $H$ arises from an isomorphism between $\frak{A}$ and $\frak{B}$; if $\frak{A}$ is nowhere rigid, $\Aut\frak{A}$ has no outer automorphisms; applications to localizable measure algebras.} \section{385}{Entropy}{21.10.03}{303}{} {Entropy of a partition of unity in a probability algebra; conditional entropy; entropy of a measure-preserving homomorphism; calculation of entropy (Kolmogorov-Sina\v{\i} theorem); Bernoulli shifts; isomorphic homomorphisms and conjugacy classes in $\Aut_{\bar\mu}\frak{A}$; almost isomorphic \imp\ functions.} \section{386}{More about entropy}{20.8.15}{316}{} {The Halmos-Rokhlin-Kakutani lemma; the Shannon-McMillan-Breiman theorem; the Csisz\'ar-Kullback inequality; various lemmas.} \section{387}{Ornstein's theorem}{9.3.16}{326}{} {Bernoulli partitions; finding Bernoulli partitions with elements of given measure (Sina\v{\i}'s theorem); adjusting Bernoulli partitions; Ornstein's theorem (Bernoulli shifts of the same finite entropy are isomorphic); Ornstein's and Sina\v{\i}'s theorems in the case of infinite entropy.} \section{388}{Dye's theorem}{14.1.04}{347}{} {Orbits of \imp\ functions; von Neumann transformations; von Neumann transformations generating a given full subgroup; classification of full subgroups generated by a single automorphism.} \wheader{}{10}{4}{4}{100pt} Chapter 39: Measurable algebras \chapintrosection{17.11.10}{360}{} \section{391}{Kelley's theorem}{5.9.07}{360}{} {Measurable algebras; strictly positive additive functionals and weak $(\sigma,\infty)$-distributivity; additive functionals subordinate to or dominating a given functional; intersection numbers; existence of strictly positive additive functionals.} \section{392}{Submeasures}{11.2.08}{365}{} {Submeasures; exhaustive, uniformly exhaustive and Maharam submeasures; the Kalton-Roberts theorem (a strictly positive uniformly exhaustive submeasure provides a strictly positive additive functional); strictly positive submeasures, associated metrics and metric completions of algebras; products of submeasures.} \section{393}{Maharam algebras}{11.5.08}{372}{} {Maharam submeasures; Maharam algebras; topologies on Boolean algebras; order-sequential topologies; characterizations of Maharam algebras.} \section{394}{Talagrand's example}{13.6.11}{382}{} {An exhaustive submeasure which is not uniformly exhaustive; a non-measurable Maharam algebra; control measures.} \section{395}{Kawada's theorem}{15.6.08}{393}{}{Full local semigroups; $\tau$-equidecomposability; fully non-paradoxical subgroups of $\Aut\frak{A}$; \hbox{$\low{b:a}$} and $\high{b:a}$; invariant additive functions from $\frak{A}$ to $L^{\infty}(\frak{C})$, where $\frak{C}$ is the fixed-point subalgebra of a group; invariant additive functionals and measures; ergodic fully non-paradoxical groups.} \section{396}{The Hajian-Ito theorem}{15.8.08}{406}{} {Invariant measures on measurable algebras; weakly wandering elements.} \wheader{}{10}{4}{4}{100pt} Appendix to Volume 3 \chapintrosection{13.3.08}{409}{} \section{3A1}{Set theory}{31.10.07}{409}{} {Calculation of cardinalities; cofinal sets, cofinalities; notes on the use of Zorn's Lemma; the natural numbers as finite ordinals; lattice homomorphisms; the Marriage Lemma.} \section{3A2}{Rings}{22.11.07}{411}{} {Rings; subrings, ideals, homomorphisms, quotient rings, the First Isomorphism Theorem; products.} \section{3A3}{General topology}{14.12.07}{414}{} {Hausdorff, regular, completely regular, zero-dimensional, extremally disconnected, compact and locally compact spaces; continuous functions; dense subsets; meager sets; Baire's theorem for locally compact spaces; products; Tychonoff's theorem; the usual topologies on $\{0,1\}^I$, $\BbbR^I$; cluster points of filters; topology bases; uniform convergence of sequences of functions; one-point compactifications; topologies defined from sequential convergences.} \section{3A4}{Uniformities}{30.1.08}{418}{} {Uniform spaces; and pseudometrics; uniform continuity; subspaces; product uniformities; Cauchy filters and completeness; extending uniformly continuous functions; completions.} \section{3A5}{Normed spaces}{22.5.11}{420}{} {The Hahn-Banach theorem in analytic and geometric forms; cones and convex sets; weak and weak* topologies; reflexive spaces; Uniform Boundedness Theorem; strong operator topologies; completions; normed algebras; compact linear operators; Hilbert spaces; bounded sets in linear topological spaces.} \section{3A6}{Groups}{6.8.08}{423}{} {Involutions; inner and outer automorphisms; normal subgroups.} \medskip Concordance to part II \pagereference{424}{} \medskip References for Volume 3 \vtmpb{9.4.05}\pagereference{425}{} \medskip Index to Volumes 1-3 \qquad Principal topics and results \pagereference{429}{} \qquad General index \pagereference{436}{} %468 pp \bigskip \noindent Volume 4: Topological Measure Spaces \medskip \noindent Part I \wheader{}{10}{4}{4}{100pt} Introduction to Volume 4 \vtmpb{14.12.06}\pagereference{12}{} \wheader{}{10}{4}{4}{100pt} Chapter 41: Topologies and measures I \chapintrosection{17.4.10}{14}{} \section{411}{Definitions}{31.12.08}{14}{} {Topological, inner regular, $\tau$-additive, outer regular, locally finite, effectively locally finite, quasi-Radon, Radon, completion regular, Baire, Borel and strictly positive measures; measurable and almost continuous functions; self-supporting sets and supports of measures; Stone spaces; Dieudonn\'e's measure.} \section{412}{Inner regularity}{31.1.05}{20}{} {Exhaustion; Baire measures; Borel measures on metrizable spaces; completions and c.l.d.\ versions; complete locally determined spaces; \imp\ functions; subspaces; indefinite-integral measures; products; outer regularity.} \section{413}{Inner measure constructions}{25.2.05}{31}{} {Inner measures; constructing a measure from an inner measure; the inner measure defined by a measure; complete locally determined spaces; extension of functionals to measures; countably compact classes; constructing measures dominating given functionals.} \section{414}{$\tau$-additivity}{26.1.10}{50}{} {Semi-continuous functions; supports; strict localizability; subspace measures; regular topologies; density topologies; lifting topologies.} \allowmorestretch{468}{ \section{415}{Quasi-Radon measure spaces}{7.4.05}{57}{} {Strict localizability; subspaces; regular topologies; hereditarily Lindel\"of spaces; products of separable metrizable spaces; comparison and specification of quasi-Radon measures; construction of quasi-Radon measures extending given functionals; indefinite-integral measures; $L^p$ spaces; Stone spaces.} }%end of allowmorestretch \section{416}{Radon measure spaces}{24.6.05}{71}{} {Radon and quasi-Radon measures; specification of Radon measures; c.l.d.\ versions of Borel measures; locally compact topologies; constructions of Radon measures extending or dominating given functionals; additive functionals on Boolean algebras and Radon measures on Stone spaces; subspaces; products; Stone spaces of measure algebras; compact and perfect measures; representation of homomorphisms of measure algebras.} \section{417}{$\tau$-additive product measures}{9.3.10}{85}{} {The product of two effectively locally finite $\tau$-additive measures; the product of many $\tau$-additive probability measures; Fubini's theorem; generalized associative law; measures on subproducts as image measures; products of strictly positive measures; quasi-Radon and Radon product measures; when `ordinary' product measures are $\tau$-additive; continuous functions and Baire $\sigma$-algebras in product spaces.} \section{418}{Measurable functions and almost continuous functions} {19.8.05}{106}{} {Measurable functions; into (separable) metrizable spaces; and image measures; almost continuous functions; continuity, measurability, image measures; expressing Radon measures as images of Radon measures; Prokhorov's theorem on projective limits of Radon measures; representing measurable functions into $L^0$ spaces.} \section{419}{Examples}{2.12.05}{121}{} {A nearly quasi-Radon measure; a Radon measure space in which the Borel sets are inadequate; a nearly Radon measure; the Stone space of the Lebesgue measure algebra; measures with domain $\Cal{P}\omega_1$; notes on Lebesgue measure; the split interval.} \wheader{}{10}{4}{4}{100pt} Chapter 42: Descriptive set theory \chapintrosection{30.9.08}{133}{} \section{421}{Souslin's operation}{14.12.07}{133}{} {Souslin's operation; is idempotent; as a projection operator; Souslin-F sets; *constituents.} \section{422}{K-analytic spaces}{4.12.04}{142}{} {Usco-compact relations; K-analytic sets; and Souslin-F sets; *First Separation Theorem.} \section{423}{Analytic spaces}{13.8.05}{148}{} {Analytic spaces; are K-analytic spaces with countable networks; Souslin-F sets; Borel measurable functions; injective images of Polish spaces; non-Borel analytic sets; a von Neumann-Jankow selection theorem; *constituents of coanalytic sets.} \section{424}{Standard Borel spaces}{21.3.08}{158}{} {Basic properties; isomorphism types; subspaces; Borel measurable actions of Polish groups.} \section{*425}{Realization of automorphisms}{9.8.13}{164}{} {Extending group actions; T\"ornquist's theorem.} \wheader{}{10}{4}{4}{100pt} Chapter 43: Topologies and measures II \chapintrosection{21.8.15}{173}{} \section{431}{Souslin's operation}{4.8.15}{173}{} {The domain of a complete locally determined measure is closed under Souslin's operation; the kernel of a Souslin scheme is approximable from within; Baire-property algebras; $\omega_1$-saturated ideals.} \section{432}{K-analytic spaces}{2.10.13}{177}{} {Topological measures on K-analytic spaces; extensions to Radon measures; expressing Radon measures as images of Radon measures; Choquet capacities.} \section{433}{Analytic spaces}{27.6.10}{182}{} {Measures on spaces with countable networks; inner regularity of Borel measures; expressing Radon measures as images of Radon measures; measurable and almost continuous functions; the von Neumann-Jankow selection theorem; products; extension of measures on $\sigma$-subalgebras; standard Borel spaces.} \section{434}{Borel measures}{18.1.14}{186}{} {Classification of Borel measures; Radon spaces; universally measurable sets and functions; Borel-measure-compact, Borel-measure-complete and pre-Radon spaces; countable compactness and countable tightness; quasi-dyadic spaces and completion regular measures; first-countable spaces and Borel product measures.} \section{435}{Baire measures}{16.8.08}{204}{} {Classification of Baire measures; extension of Baire measures to Borel measures (Ma\v{r}\'{\i}k's theorem); measure-compact spaces.} \section{436}{Representation of linear functionals}{9.5.11}{209}{} {Smooth and sequentially smooth linear functionals; measures and sequentially smooth functionals; Baire measures; sequential spaces and products of Baire measures; quasi-Radon measures and smooth functionals; locally compact spaces and Radon measures.} \section{437}{Spaces of measures}{5.11.12}{219}{} {Smooth and sequentially smooth duals; signed measures; embedding spaces of measurable functions in the bidual of $C_b(X)$; vague and narrow topologies; product measures; extreme points; uniform tightness; total variation metric, Kantorovich-Rubinshtein metric; invariant probability measures; Prokhorov spaces.} \section{438}{Measure-free cardinals}{13.12.06}{244}{} {Measure-free cardinals; point-finite families of sets with measurable unions; measurable functions into metrizable spaces; Radon and measure-compact metric spaces; metacompact spaces; hereditarily weakly $\theta$-refinable spaces; when $\frak{c}$ is measure-free.} \section{439}{Examples}{7.7.14}{259}{} {Measures on $[0,1]$ not extending to Borel measures; universally negligible sets; Hausdorff measures are rarely semi-finite; a smooth linear functional not expressible as an integral; a first-countable non-Radon space; Baire measures not extending to Borel measures; $\BbbN^{\frak{c}}$ is not Borel-measure-compact; the Sorgenfrey line; $\Bbb{Q}$ is not a Prokhorov space.} \wheader{}{10}{4}{4}{100pt} Chapter 44: Topological groups \chapintrosection{18.4.08}{274}{} \section{441}{Invariant measures on locally compact spaces}{3.1.06}{274}{} {Measures invariant under homeomorphisms; Haar measures; measures invariant under isometries.} \section{442}{Uniqueness of Haar measure}{21.3.07}{283}{} {Two (left) Haar measures are multiples of each other; left and right Haar measures; Haar measurable and Haar negligible sets; the modular function of a group; formulae for ${\int}f(x^{-1})dx$, ${\int}f(xy)dx$.} \section{443}{Further properties of Haar measure}{14.1.13}{289}{} {The Haar measure algebra of a group carrying Haar measures; actions of the group on the Haar measure algebra; locally compact groups; actions of the group on $L^0$ and $L^p$; the bilateral uniformity; Borel sets are adequate; completing the group; expressing an arbitrary Haar measure in terms of a Haar measure on a locally compact group; completion regularity of Haar measure; invariant measures on the set of left cosets of a closed subgroup of a locally compact group; modular functions of subgroups and quotient groups; transitive actions of compact groups on compact spaces.} \section{444}{Convolutions}{23.7.07}{314}{} {Convolutions of quasi-Radon measures; the Banach algebra of signed $\tau$-additive measures; continuous actions and corresponding actions on $L^0(\nu)$ for an arbitrary quasi-Radon measure $\nu$; convolutions of measures and functions; indefinite-integral measures over a Haar measure $\mu$; convolutions of functions; $L^p(\mu)$; approximate identities; convolution in $L^2(\mu)$.} \section{445}{The duality theorem}{20.3.08}{336}{} {Dual groups; Fourier-Stieltjes transforms; Fourier transforms; identifying the dual group with the maximal ideal space of $L^1$; the topology of the dual group; positive definite functions; Bochner's theorem; the Inversion Theorem; the Plancherel Theorem; the Duality Theorem.} \section{446}{The structure of locally compact groups}{8.10.13}{357}{} {Finite-dimensional representations separate the points of a compact group; groups with no small subgroups have $B$-sequences; chains of subgroups.} \ifdim\pagewidth>467pt\fontdimen3\tenrm=2pt\fi \section{447}{Translation-invariant liftings}{7.1.10}{371}{} {Translation-invariant liftings and lower densities; Vitali's theorem and a density theorem for groups with $B$-sequences; Haar measures have transla\discretionary{}{-}{}tion-{\vthsp}invariant liftings.} \fontdimen3\tenrm=1.67pt \section{448}{Polish group actions} {12.4.13}{382}{} {Countably full local semigroups of $\Aut\frak{A}$; $\sigma$-equidecomposability; countably non-paradoxical groups; $G$-invariant additive functions from $\frak{A}$ to $L^{\infty}(\frak{C})$; measures invariant under Polish group actions (the Nadkarni-Becker-Kechris theorem); measurable liftings of $L^0$; the Borel structure of $L^0$; representing a Borel measurable action on a measure algebra by a Borel measurable action on a Polish space (Mackey's theorem).} \section{449}{Amenable groups}{13.6.13}{397}{} {Amenable groups; permanence properties; the greatest ambit of a topological group; locally compact amenable groups; Tarski's theorem; discrete amenable groups; isometry-invariant extensions of Lebesgue measure.} \wheader{}{10}{2}{2}{100pt} Chapter 45: Perfect measures, disintegrations and processes \chapintrosection{5.6.09}{420}{} \section{451}{Perfect, compact and countably compact measures}{8.11.07}{421}{} {Basic properties of the three classes; subspaces, completions, c.l.d.\ versions, products; measurable functions from compact measure spaces to metrizable spaces; *weakly $\alpha$-favourable spaces.} \section{452}{Integration and disintegration of measures}{6.11.08}{434}{} {Integrating families of measures; $\tau$-additive and Radon measures; disintegrations and regular conditional probabilities; disintegrating countably compact measures; disintegrating Radon measures; *images of countably compact measures.} \section{453}{Strong liftings}{22.3.10}{449}{} {Strong and almost strong liftings; existence; on product spaces; disintegrations of Radon measures over spaces with almost strong liftings; Stone spaces; Losert's example.} \section{454}{Measures on product spaces}{6.12.05}{461}{} {Perfect, compact and countably compact measures on product spaces; extension of finitely additive functions with perfect countably additive marginals; Kolmogorov's extension theorem; measures defined from conditional distributions; distributions of random processes; measures on $C(T)$ for Polish $T$; completion regular product measures.} \section{455}{Markov and L\'evy processes} {18.1.09}{470}{} {Realization of a Markov process with given conditional distributions; the Markov property for stopping times taking countably many values -- disintegrations and conditional expectations; Radon conditional distributions; narrowly continuous and uniformly time-continuous systems of conditional distributions; \cadlag{} and \callal{} functions; extending the distribution of a process to a Radon measure; when the subspace measure on the \cadlag{} functions is quasi-Radon; general stopping times, hitting times; the strong Markov property; independent increments, L\'evy processes; expressing the strong Markov property with an \imp\ function.} \section{456}{Gaussian distributions}{19.5.10}{510}{} {Gaussian distributions and processes; covariance matrices, correlation and independence; supports; universal Gaussian distributions; cluster sets of $n$-dimensional processes; $\tau$-additivity.} \section{457}{Simultaneous extension of measures}{18.1.13}{527}{} {Extending families of finitely additive functionals; Strassen's theorem; extending families of measures; examples; the Wasserstein metric.} \section{458}{Relative independence and relative products}{16.1.07}{543}{} {Relatively independent algebras of measurable sets; relative distributions and relatively independent random variables; relatively independent subalgebras of a probability algebra; relative free products of probability algebras; relative products of probability spaces; existence of relative products.} \section{459}{Symmetric measures and exchangeable random variables}{7.12.10}{562}{} {Exchangeable families of \imp\ functions; De Finetti's theorem; countably compact symmetric measures on product spaces disintegrate into product measures; symmetric quasi-Radon measures; other actions of symmetric groups.} \wheader{}{10}{4}{4}{20pt} Concordance to Part I \pagereference{576}{} %577 pages \medskip \noindent Part II \wheader{}{10}{4}{4}{100pt} Chapter 46: Pointwise compact sets of measurable functions \chapintrosection{26.8.13}{8}{} \section{461}{Barycenters and Choquet's theorem}{9.7.08}{8}{} {Barycenters; elementary properties; sufficient conditions for existence; closed convex hulls of compact sets; \Krein's theorem; existence and uniqueness of measures on sets of extreme points; ergodic functions and extreme measures.} \section{462}{Pointwise compact sets of continuous functions}{30.6.07}{20}{} {Angelic spaces; the topology of pointwise convergence on $C(X)$; weak convergence and weakly compact sets in $C_0(X)$; Radon measures on $C(X)$; separately continuous functions; convex hulls.} \section{463}{$\frak{T}_p$ and $\frak{T}_m$}{1.2.13}{26}{} {Pointwise convergence and convergence in measure on spaces of measurable functions; compact and sequentially compact sets; perfect measures and Fremlin's Alternative; separately continuous functions.} \section{464}{Talagrand's measure}{25.5.13}{36}{} {The usual measure on $\Cal{P}I$; the intersection of a sequence of non-measurable filters; Talagrand's measure; the $L$-space of additive functionals on $\Cal{P}I$; measurable and purely non-measurable functionals.} \ifdim\pagewidth>467pt\fontdimen3\tenrm=2pt\fi \section{465}{Stable sets}{28.1.06}{48}{} {Stable sets of functions; elementary properties; pointwise compactness; pointwise convergence and convergence in measure; a law of large numbers; stable sets and uniform convergence in the strong law of large numbers; convex hulls; stable sets in $L^0$ and $L^1$; *R-stable sets.} \fontdimen3\tenrm=1.67pt \section{466}{Measures on linear topological spaces}{2.8.13}{76}{} {Quasi-Radon measures for weak and strong topologies; Kadec norms; constructing weak-Borel measures; characteristic functions of measures on locally convex spaces; universally measurable linear operators; Gaussian measures on linear topological spaces.} \section{*467}{Locally uniformly rotund norms}{13.1.10}{85}{} {Locally uniformly rotund norms; separable normed spaces; long sequences of projections; K-countably determined spaces; weakly compactly generated spaces; Banach lattices with order-continuous norms; Eberlein compacta and Schachermeyer's theorem.} \wheader{}{10}{4}{4}{100pt} Chapter 47: Geometric measure theory \chapintrosection{8.4.13}{97}{} \section{471}{Hausdorff measures}{10.2.16}{97}{} {Metric outer measures; Increasing Sets Lemma; analytic spaces; inner regularity; Vitali's theorem and a density theorem; Howroyd's theorem.} \section{472}{Besicovitch's Density Theorem}{22.3.11}{113}{} {Besicovitch's Covering Lemma; Besicovitch's Density Theorem; *a maximal theorem.} \section{473}{Poincar\'e's inequality}{25.7.11}{120}{} {Differentiable and Lipschitz functions; smoothing by convolution; the Gagliardo-\vthsp{}Nirenberg-\vthsp{}Sobolev inequality; Poin\discretionary{-}{}{}car\'e's inequality for balls.} \section{474}{The distributional perimeter}{17.11.12}{132}{} {The divergence of a vector field; sets with locally finite perimeter, perimeter measures and outward-normal functions; the reduced boundary; invariance under isometries; isoperimetric inequalities; Federer exterior normals; the Compactness Theorem.} \ifdim\pagewidth>467pt\fontdimen3\tenrm=2pt\fi \section{475}{The essential boundary}{24.1.13}{153}{} {Essential interior, closure and boundary; the reduced boundary, the essential boundary and perimeter measures; characterizing sets with locally finite perimeter; the Divergence Theorem; calculating perimeters from cross-sectional counts, and an integral-geometric formula; Cauchy's Perimeter Theorem; the Isoperimetric Theorem for convex sets.} \fontdimen3\tenrm=1.67pt \section{476}{Concentration of measure}{10.11.07}{176}{} {Vietoris and Fell topologies; concentration by partial reflection; concentration of measure in $\eightBbb{R}^r$; the Isoperimetric Theorem; concentration of measure on spheres.} \section{477}{Brownian motion}{2.1.10}{185}{} {Brownian motion as a stochastic process; Wiener measure on $C(\coint{0,\infty})_0$; *as a limit of random walks; Brownian motion in $\eightBbb{R}^r$; invariant transformations of Wiener measure on $C(\coint{0,\infty};\eightBbb{R}^r)_0$; Wiener measure is strictly positive; the strong Markov property; hitting times; almost every Brownian path is nowhere differentiable; almost every Brownian path has zero two-dimensional Hausdorff measure.} \section{478}{Harmonic functions}{4.6.09}{210}{} {Harmonic and superharmonic functions; a maximal principle; $f$ is superharmonic iff $\nabla^2f\le0$; the Poisson kernel and harmonic functions with given values on a sphere; smoothing by convolution; Brownian motion and Dynkin's formula; Brownian motion and superharmonic functions; recurrence and divergence of Brownian motion; harmonic measures and Dirichlet's problem; disintegrating harmonic measures over intermediate boundaries; hitting probabilities.} \section{479}{Newtonian capacity}{15.2.10}{235}{} {Defining Newtonian capacity from Brownian hitting probabilities, and equilibrium measures from harmonic measures; submodularity and sequential order-continuity; extending Newtonian capacity to Choquet-Newton capacity; Newtonian potential and energy of a Radon measure; Riesz kernels and their Fourier transforms; energy and $(r-1)$-potentials; alternative definitions of capacity and equilibrium measures; analytic sets of finite capacity; polar sets; general sets of finite capacity; Brownian hitting probabilities and equilibrium potentials; Hausdorff measure; self-intersecting Brownian paths; a discontinuous equilibrium potential; yet another definition of Newtonian capacity; capacity and volume; a measure on the set of closed subsets of $\eightBbb{R}^r$.} \wheader{}{10}{4}{4}{100pt} Chapter 48: Gauge integrals \chapintrosection{9.5.11}{277}{} \section{481}{Tagged partitions}{4.9.09}{277}{} {Tagged partitions and Riemann sums; gauge integrals; gauges; residual sets; subdivisions; examples (the Riemann integral, the Henstock integral, the symmetric Riemann-complete integral, the McShane integral, box products, the approximately continuous Henstock integral).} \ifdim\pagewidth>467pt\fontdimen3\tenrm=2pt\fi \section{482}{General theory}{11.5.10}{287}{} {Saks-Henstock lemma; when gauge-{\vthsp}integrable functions are measurable; when integrable functions are gauge-integrable; $I_{\nu}(f\times{\chi}H)$; improper integrals; integrating derivatives; B.Levi's theorem; Fubini's theorem.} \fontdimen3\tenrm=1.67pt \section{483}{The Henstock integral}{6.9.10}{301}{} {The Henstock and Lebesgue integrals; indefinite Henstock integrals; Saks-Henstock lemma; Fundamental Theorem of Calculus; the Perron integral; $\|f\|_H$ and $HL^1$; AC$_*$ and ACG$_*$ functions.} \section{484}{The Pfeffer integral}{21.1.10}{317}{} {The Tamanini-Giacomelli theorem; %or Congedo-Tamanini a family of tagged-partition structures; the Pfeffer integral; the Saks-Henstock indefinite integral of a Pfeffer integrable function; Pfeffer's Divergence Theorem; differentiating the indefinite integral; invariance under lipeomorphisms.} \wheader{}{10}{4}{4}{100pt} Chapter 49: Further topics \chapintrosection{31.8.09}{335}{} \section{491}{Equidistributed sequences}{14.8.08}{335}{} {The asymptotic density ideal $\Cal{Z}$; equidistributed sequences; when equidistributed sequences exist; $\frak{Z}=\Cal{P}\eightBbb{N}/\Cal{Z}$; effectively regular measures; equidistributed sequences and induced embeddings of measure algebras in $\frak{Z}$.} \section{492}{Combinatorial concentration of measure}{30.12.06}{353}{} {The Hamming metric; concentration of measure in product spaces; concentration of measure in permutation groups.} \section{493}{Extremely amenable groups}{4.1.13}{360}{} {Extremely amenable groups; concentrating additive functionals; measure algebras under $\Bsymmdiff$; $L^0$; isometry groups of spheres in inner product spaces; locally compact groups.} \section{494}{Groups of measure-preserving automorphisms} {17.5.13}{368}{} {Weak and uniform topologies on $\AmuA$; a weakly mixing automorphism which is not mixing; full subgroups and fixed-point subalgebras; extreme amenability; automatic continuity; algebraic cofinality.} \section{495}{Poisson point processes}{20.12.08}{399}{} {Poisson distributions; Poisson point processes; disintegrations; transforming disjointness into stochastic independence; representing Poisson point processes by Radon measures; exponential distributions and Poisson point processes on $\coint{0,\infty}$.} \section{496}{Maharam submeasures}{27.5.09}{421}{} {Submeasures; totally finite Radon submeasures; Souslin's operation; (K-)analytic spaces; product submeasures.} \section{497}{Tao's proof of Szemer\'edi's theorem}{7.12.10}{430}{} {$\Tau$-removable intersections; and relative independence; permutation-invariant measures on $\Cal{P}([I]^{<\omega})$; and $\Tau$-removable intersections; the Hypergraph Removal Lemma; Szemer\'edi's theorem; a multiple recurrence theorem.} \section{498}{Cubes in product spaces}{4.9.08}{442}{} {Subsets of measure algebras with non-zero infima; product sets included in given sets of positive measure.} \wheader{}{10}{4}{4}{100pt} Appendix to Volume 4 \chapintrosection{3.9.13}{445}{} \section{4A1}{Set theory}{27.1.13}{445}{} {Cardinals; closed cofinal sets and stationary sets; $\Delta$-system lemma; free sets; Ramsey's theorem; the Marriage Lemma again; filters; normal ultrafilters; Ostaszewski's $\clubsuit$; the size of $\sigma$-algebras.} \section{4A2}{General topology}{21.4.13}{449}{} {Glossary; general constructions; F$_{\sigma}$, G$_{\delta}$, zero and cozero sets; weight; countable chain condition; separation axioms; compact and locally compact spaces; Lindel\"of spaces; Stone-\v{C}ech compactifications; uniform spaces; first-{\vthsp}countable, sequential, countably tight, metrizable spaces; countable networks; second-countable spaces; separable metrizable spaces; Polish spaces; order topologies; Vietoris and Fell topologies.} \section{4A3}{Topological $\sigma$-algebras}{11.10.07}{472}{} {Borel $\sigma$-algebras; measurable functions; hereditarily Lindel\"of spaces; second-countable spaces; Polish spaces; $\omega_1$; Baire $\sigma$-algebras; product spaces; compact spaces; Baire-property algebras; cylindrical $\sigma$-algebras; spaces of \cadlag{} functions.} \section{4A4}{Locally convex spaces}{19.6.13}{482}{} {Linear topological spaces; locally convex spaces; Hahn-Banach theorem; normed spaces; inner product spaces; max-flow min-cut theorem.} \section{4A5}{Topological groups}{4.8.13}{488}{} {Group actions; topological groups; uniformities; quotient groups; metrizable groups.} \section{4A6}{Banach algebras}{8.12.10}{495}{} {Stone-Weierstrass theorem (fourth form); multiplicative linear functionals; spectral radius; invertible elements; exponentiation; Arens multiplication.} \section{4A7}{`Later editions only'}{5.10.13}{499}{} {Items recently interpolated into other volumes.} \wheader{}{10}{2}{2}{100pt} Concordance to Part II \pagereference{499}{} \medskip References for Volume 4 \vtmpb{9.2.13}\pagereference{500}{} \medskip Index to Volumes 1-4 \qquad Principal topics and results \pagereference{507}{} \qquad General index \pagereference{519}{} %573 pages \bigskip \noindent Volume 5: Set-Theoretic Measure Theory \medskip \noindent Part I \wheader{}{10}{4}{4}{100pt} Introduction to Volume 5 \vtmpb{9.1.15}\pagereference{}{10} \wheader{}{10}{4}{4}{100pt} Chapter 51: Cardinal functions %page-ends checked 8.1.15 \chapintrosection{3.1.15}{}{12} \section{511}{Definitions}{10.10.13}{}{12} {Cardinal functions of partially ordered sets, topological spaces, Boolean algebras and measures; precalibers; ideals of sets.} \section{512}{Galois-Tukey connections}{27.11.13}{}{20} {Supported relations; Galois-Tukey connections; covering numbers, additivity, saturation, linking numbers; simple products; sequential composition of supported relations.} \section{513}{Partially ordered sets}{23.2.14}{}{25} {Saturation and the Erd\H{o}s-Tarski theorem; cofinalities of cardinal functions; Tukey functions; Tukey equivalence of directed sets; $\sigma$-additivities; *metrizably compactly based directed sets; *measurable Tukey functions.} \section{514}{Boolean algebras}{16.5.14}{}{36} {Stone spaces; cardinal functions of Boolean algebras; order-preserving functions of Boolean algebras; regular open algebras; regular open algebras of partially ordered sets; finite-support products.} \section{515}{The Balcar-Fran\v{e}k theorem}{29.8.14}{}{54} {Boolean-independent sets; free subalgebras; refining systems; the 0Balcar-Fran\v{e}k theorem; the Pierce-Koppelberg theorem; regular open algebras of powers of $\{0,1\}$.} \section{516}{Precalibers}{9.10.14}{}{61} {Precalibers of supported relations; and Galois-Tukey connections; partially ordered sets, topological spaces and Boolean algebras; saturation and linking numbers; saturation of product spaces.} \section{517}{Martin numbers}{14.11.14}{}{68} {Characterizations of $\frak{m}(P)$; regular open algebras, Stone spaces and Nov\'ak numbers; precalibers, saturation and weak distributivity; $\frak{m}$, $\frakmctbl$, $\frak{p}$ and $\frak{m}_{\text{K}}$.} \section{518}{Freese-Nation numbers}{24.12.14}{}{78} {Freese-Nation numbers of partially ordered sets; Boolean algebras; upper and lower bounds for $\FN(\frak{A})$ under special axioms; tight filtrations and Geschke systems; large algebras are not tightly filtered.} \wheader{}{10}{4}{4}{100pt} Chapter 52: Cardinal functions of measure theory %page-ends checked 8.1.15 \chapintrosection{7.10.13}{}{90} \section{521}{Basic theory}{3.3.14}{}{90} {$\add\mu$ and $\add\Cal{N}(\mu)$; measure algebras and function spaces; the topological density of a measure algebra; shrinking numbers; $\pi(\mu)$; subspace measures, direct sums, image measures, products; perfect measures, compact measures; complete locally determined measure spaces and strict localizability; magnitudes; bounds on the Maharam type of a measure; countably separated spaces; measurable additive functionals on $\Cal{P}I$.} \section{522}{Cicho\'n's diagram}{31.3.10}{}{104} {The cardinals $\frak{b}$ and $\frak{d}$; inequalities linking them with the additivity, cofinality, uniformity and covering numbers of measure and category in the real line; the localization relation; $\frakmctbl$ and other Martin numbers; $\FN(\Cal{P}\eightBbb{N})$; cofinalities of the cardinals.} \section{523}{The measure of $\{0,1\}^I$}{23.6.10}{}{121} {The additivity, covering number, uniformity, shrinking number and cofinality of the usual measure on $\{0,1\}^I$; Kraszewski's theorems; what happens with GCH.} \section{524}{Radon measures}{29.9.10}{}{128} {The additivity, covering number, uniformity and cofinality of a Radon measure; $\ell^1(\kappa)$ and localization; cardinal functions of measurable algebras; countably compact and quasi-Radon measures.} \section{525}{Precalibers of measure algebras}{11.9.13}{}{145} {Precalibers of measurable algebras; measure-precalibers of probability algebras; (quasi-\nobreak)Radon measure spaces; under GCH; precaliber triples $(\kappa,\kappa,k)$.} \section{526}{Asymptotic density zero}{24.1.14}{}{154} {$\Cal{Z}$ is metrizably compactly based; $\NN\prT\Cal{Z}\prT\ell^1\prGT\NN\ltimes\Cal{Z}$; cardinal functions of $\Cal{Z}$; meager sets and nowhere dense sets; sets with negligible closures; $\CalNwd\not\prT\Cal{Z}$ and $\Cal{Z}\not\prT\CalNwd$.} \section{527}{Skew products of ideals}{10.1.10}{}{167} {$\Cal{N}\ltimes_{\Cal{B}}\Cal{N}$ and Fubini's theorem; $\Cal{M}\ltimes_{\Cal{B}}\Cal{M}$ and the Kuratowski-Ulam theorem; $\Cal{M}\ltimes_{\Cal{B}}\Cal{N}$; $\Cal{N}\ltimes_{\Cal{B}}\Cal{M}$; harmless Boolean algebras.} \section{528}{Amoeba algebras}{10.2.11}{}{178} {Amoeba algebras; variable-measure amoeba algebras; isomorphic amoeba algebras; regular embeddings of amoeba algebras; localization posets; Martin numbers and other cardinal functions; algebras with countable Maharam type.} \section{529}{Further partially ordered sets of analysis}{26.5.11}{}{200} {$L^p$ and $L^0$; $L$-spaces; the localization poset and the regular open algebra of $\{0,1\}^{\frakc}$; the Nov\'ak numbers $n(\{0,1\}^I)$; the reaping numbers $\frak{r}(\omega_1,\lambda)$.} \wheader{}{10}{4}{4}{100pt} Chapter 53: Topologies and measures III \chapintrosection{30.8.14}{}{208} \section{531}{Maharam types of Radon measures}{19.2.11}{}{208} {Topological and measure-theoretic cardinal functions; the set $\MahR(X)$ of Maharam types of homogeneous Radon measures on $X$; $\MahR(X)$, precalibers and continuous surjections onto $[0,1]^{\kappa}$; $\MahR(X)$ and $\chi(X)$; a perfectly normal hereditarily separable space under CH; when $\frak{m}_{\text{K}}>\omega_1$.} \section{532}{Completion regular measures on $\{0,1\}^I$} {1.6.13}{}{226} {The set $\MahcrR(X)$ of Maharam types of homogeneous completion regular Radon measures on $X$; products of quasi-dyadic spaces; convexity of the relation `$\lambda\in\MahcrR(\{0,1\}^{\kappa})$'; the measure algebra of $\{0,1\}^{\lambda}$; $\frak{d}$, $\cov\Cal{N}$, $\FN(\Cal{P}\Bbb{N})$, $\add\Cal{N}$ and the case $\lambda=\omega$; $\square$, Chang's conjecture and the case $\cf\lambda=\omega$.} \section{533}{Special topics}{4.1.14}{}{236} {$\add\Cal{N}$ and (quasi-)Radon measures of countable Maharam type; uniformly regular measures; when $\eightBbb{R}^{\kappa}$ is measure-compact.} \section{534}{Hausdorff measures and strong measure zero} {1.8.10}{}{243} {Cardinal functions of Hausdorff measures; strong measure zero in uniform spaces; Rothberger's property; $\sigma$-compact groups; $\non\CalSmz$, $\add\CalSmz$; $\CalSmz$-equivalence; uncountable sets with strong measure zero.} \section{535}{Liftings}{1.6.11}{}{257} {Liftings of non-complete measure spaces; Baire liftings for usual measures on $\{0,1\}^{\kappa}$; tightly $\omega_1$-filtered measure algebras; Mokobodzki's theorems; strong Borel liftings; Borel liftings for Radon measures on metrizable spaces; linear liftings; problems.} \section{536}{Alexandra Bellow's problem}{20.2.12}{}{269} {The problem; consequences of a negative solution.} \section{537}{Sierpi\'nski sets, shrinking numbers and strong Fubini theorems}{12.8.13}{}{273} {Sierpi\'nski and strongly Sierpi\'nski sets; entangled totally ordered sets; non-ccc products; scalarly measurable functions; repeated integrals of separately measurable functions; changing the order of integration in multiply repeated integrals; $\shr^+$, $\cov$ and repeated upper and lower integrals.} \section{538}{Filters and limits}{18.2.14}{}{287} {Filters on $\Bbb{N}$; the Rudin-Keisler ordering; products and iterated products; Ramsey ultrafilters; measure-centering ultrafilters; extending perfect measures with measure-centering ultrafilters; Benedikt's theorem; measure-converging filters; the Fatou property; medial functionals and limits.} \section{539}{Maharam submeasures}{24.5.14}{}{317} {Maharam algebras; Maharam-algebra topology, pre-ordered set of partitions of unity, weak distributivity, $\pi$-weight, centering number, precalibers; null ideals of Maharam submeasures; splitting reals; Quickert's ideal; Todor\v{c}evi\'c's $p$-ideal dichotomy; a consistent characterization of Maharam algebras; Souslin algebras; reflection principles; exhaustivity rank, Maharam submeasure rank.} \wheader{}{10}{4}{4}{20pt} Concordance to Part I \pagereference{}{329} %Lulu 2015: 329 pages \wheader{}{4}{2}{2}{108pt} \noindent Part II \wheader{}{10}{4}{4}{100pt} Chapter 54: Real-valued-measurable cardinals %page-ends checked 8.1.15 \chapintrosection{23.10.14}{}{7} \section{541}{Saturated ideals}{10.12.12}{}{7} {$\kappa$-saturated $\kappa^+$-additive ideals; $\kappa$-saturated $\kappa$-additive ideals; $\Tr_{\Cal{I}}(X;Y)$; normal ideals; $\kappa$-saturated normal ideals; two-valued-measurable and weakly compact cardinals; the Tarski-Solovay dichotomy; $\covSh(2^{\gamma},\kappa,\delta^+,\delta)$.} \section{542}{Quasi-measurable cardinals}{8.7.13}{}{18} {Definition and basic properties; $\omega_1$-saturated $\sigma$-ideals; and pcf theory; and cardinal arithmetic; cardinals of quotient algebras; cofinality of $[\kappa]^{<\theta}$; cofinality of product partial orders.} \section{543}{The Gitik-Shelah theorem}{11.11.13}{}{23} {Real-valued-measurable and atomlessly-measurable cardinals; Ulam's dichotomy; a Fubini inequality; Maharam types of witnessing probabilities; compact measures, inverse-measure-preserving functions and extensions of measures.} \section{544}{Measure theory with an atomlessly-measurable cardinal} {31.12.13}{}{30} {Covering numbers of null ideals; repeated integrals; measure-precalibers; functions from $[\kappa]^{<\omega}$ to null ideals; Sierpi\'nski sets; uniformities of null ideals; weakly $\Pi^1_1$-indescribable cardinals; Cicho\'n's diagram.} \section{545}{PMEA and NMA}{10.2.14}{}{41} {The product measure extension axiom; the normal measure axiom; Boolean algebras with many measurable subalgebras.} \section{546}{Power set $\sigma$-quotient algebras}{10.5.14}{}{43} {Power set $\sigma$-quotient algebras; harmless algebras and skew products of ideals; the Gitik-Shelah theorem for category algebras; the category algebra $\frak{G}_{\omega}$ of $\{0,1\}^{\omega}$; completed free products of probability algebras with $\frak{G}_{\omega}$.} \section{547}{Disjoint refinements of sequences of sets}{26.8.14}{}{60} {Refining a sequence of sets to a disjoint sequence without changing outer measures; other results on simultaneous partitions.} \wheader{}{10}{4}{4}{100pt} Chapter 55: Possible worlds \chapintrosection{7.1.15}{}{68} \section{551}{Forcing with quotient algebras}{2.12.13}{}{68} {Measurable spaces with negligibles; associated forcing notions; representing names for members of $\{0,1\}^I$; representing names for Baire sets in $\{0,1\}^I$; the usual measure on $\{0,1\}^I$; re-interpreting Baire sets in the forcing model; representing Baire measurable functions; representing measure algebras; iterated forcing; extending filters.} \section{552}{Random reals I}{29.1.14}{}{86} {Random real forcing notions; calculating $2^{\kappa}$; $\frak{b}$ and $\frak{d}$; preservation of outer measure; Sierpi\'nski sets; cardinal functions of the usual measure on $\{0,1\}^{\lambda}$; Carlson's theorem on extending measures; iterated random real forcing.} \section{553}{Random reals II}{3.5.14}{}{107} {Rothberger's property; non-scattered compact sets; Haydon's property; rapid $p$-point ultrafilters; products of ccc partially ordered sets; Aronszajn and Souslin trees; medial limits; universally measurable sets.} \section{554}{Cohen reals}{2.9.14}{}{127} {Calculating $2^{\kappa}$; Lusin sets; precaliber pairs of measure algebras; Freese-Nation numbers; Borel liftings for Lebesgue measure.} \section{555}{Solovay's construction of real-valued-measurable cardinals} {12.4.08}{}{133} {Measurable cardinals are quasi-measurable after ccc forcing, real-valued-measurable after random real forcing; Maha\discretionary{-}{}{}ram-type-homogeneity; covering number of product measure; power set $\sigma$-quotient algebras can have countable centering number or Maharam type; supercompact cardinals and the normal measure axiom.} \section{556}{Forcing with Boolean subalgebras}{3.1.15}{}{147} {Forcing names over a Boolean subalgebra; Boolean operations, ring homomorphisms; when the subalgebra is regularly embedded; upper bounds, suprema, saturation, Maharam type; quotient forcing; Dedekind completeness; $L^0$; probability algebras; relatively independent subalgebras; strong law of large numbers; Dye's theorem; Kawada's theorem; the Dedekind completion of the asymptotic density algebra.} \wheader{}{10}{4}{4}{108pt} Chapter 56: Choice and Determinacy %page-ends checked 10.1.15 \chapintrosection{3.1.15}{}{179} \section{561}{Analysis without choice}{8.9.13}{}{179} {Elementary facts; Tychonoff's theorem; Baire's theorem; Stone's theorem; Haar measure; Kakutani's representation of $L$-spaces; Hilbert space.} \section{562}{Borel codes}{20.10.13}{}{190} {Coding sets with trees; codable Borel sets; in a Polish space, a set is analytic and coanalytic iff it is a codable Borel set; resolvable sets are self-coding; codable families of codable sets; codable Borel functions, codable Borel equivalence; real-valued functions; codable families of codable functions; codable Baire sets and functions for general topological spaces.} \section{563}{Borel measures without choice}{3.12.13}{}{210} {Borel-coded measures on second-countable spaces; construction of measures; inner and outer regularity; analytic sets are universally measurable; Baire-coded measures on general topological spaces; measure algebras.} \section{564}{Integration without choice}{9.2.14}{}{220} {Integration with respect to Baire-coded measures; convergence theorems for codable sequences of functions; Riesz representation theorem; when $L^1$ is a Banach space; Radon-Nikod\'ym theorem; conditional expectations; products of measures on second-countable spaces.} \section{565}{Lebesgue measure without choice}{25.4.14}{}{236} {Construction of Lebesgue measure as a Borel-coded measure; Vitali's theorem; Fundamental Theorem of Calculus; Hausdorff measures as Borel-coded measures.} \section{566}{Countable choice}{22.8.14}{}{247} {Basic measure theory survives; exhaustion; $\sigma$-finite spaces and algebras; atomless countably additive functionals; Vitali's theorem; bounded additive functionals; infinite products without DC; topological product measures; the Loomis-Sikorski theorem; the usual measure on $\{0,1\}^{\Bbb{N}}$ and its measure algebra; weak compactness; automorphisms of measurable algebras; Baire $\sigma$-algebras; dependent choice.} \section{567}{Determinacy}{31.10.14}{}{264} {Infinite games; closed games are determined; the axiom of determinacy; AC($\Bbb{R};\omega$); universal measurability and the Baire property; automatic continuity of group homomorphisms and linear operators; countable additivity of functionals; reflexivity of $L$-spaces; $\omega_1$ is two-valued-measurable; surjections from $\Cal{P}\Bbb{N}$ onto ordinals; two-valued-measurable cardinals and determinacy in ZFC; measurability of PCA sets. } \wheader{}{10}{4}{4}{100pt} Appendix to Volume 5 %page-ends checked 8.1.15 \chapintrosection{21.11.13}{}{277} \section{5A1}{Set theory}{23.2.11}{}{277} {Ordinal and cardinal arithmetic; trees; cofinalities; $\Delta$-systems and free sets; partition calculus; transversals.} \section{5A2}{Pcf theory}{6.6.11}{}{285} {Reduced products of partially ordered sets; cofinalities of reduced products; $\covSh(\alpha,\beta,\gamma,\delta)$; $\Theta(\alpha,\gamma)$.} \section{5A3}{Forcing}{18.4.08}{}{295} {Forcing notions; forcing languages; the forcing relation; the forcing theorem; Boolean truth values; names for functions; regular open algebras; discriminating names; $L^0$ and names for real numbers; forcing with Boolean algebras; ordinals and cardinals; iterated forcing; Martin's axiom; countably closed forcings.} \section{5A4}{General topology}{15.8.13}{}{306} {Cardinal functions; compactness; Vietoris topologies; category and the Baire property; normal and paracompact spaces.} \section{5A5}{Real analysis}{3.10.13}{}{313} {Real-entire functions.} \section{5A6}{Special axioms}{4.8.10}{}{314} {GCH, \VeqL, $0^{\sharp}$ and the Covering Lemma, squares, Chang's transfer principle, Todor\v{c}evi\'c's $p$-ideal dichotomy, the filter dichotomy.} \wheader{}{10}{2}{2}{100pt} References for Volume 5 \vtmpb{6.9.13}\pagereference{}{319} %page-ends checked 10.1.15 \medskip Index to Volumes 1-5 %page-ends checked 10.1.15 \qquad Principal topics and results \pagereference{}{325} \qquad General index \pagereference{}{339} %Lulu 2015: 411 pages \bigskip \noindent Index \vtmpb{31.12.10} \bigskip \noindent Errata \medskip \quad Volume 1, 2000 edition \vtmpb{27.2.10} \quad Volume 1, 2001 printing \vtmpb{14.1.10} \quad Volume 1, 2004 printing \vtmpb{27.11.13} %\quad Volume 1, 2011 edition \vtmpb{1.12.11} \quad Volume 2, 2001 edition \vtmpb{30.12.09} \quad Volume 2, 2003 printing \vtmpb{14.6.11} \quad Volume 2, 2010 edition \vtmpb{29.11.10} \quad Volume 3, 2002 edition \vtmpb{7.8.15} \quad Volume 3, 2004 printing \vtmpb{14.8.10} %\quad Volume 3, 2012 edition \vtmpb{1.12.12} \quad Volume 4, 2003 edition \vtmpb{21.5.11} \quad Volume 4, 2006 printing \vtmpb{20.5.10} %\quad Volume 4, 2013 edition \vtmpb{1.12.13} \quad Volume 5, 2008 edition \vtmpb{21.2.11} \frnewpage %---------------------------------------------------------------% \end