Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition by J.L. Doob (PDF)

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This text is unique in accepting probability theory as an essential part of measure theory. Therefore, many examples are taken from probability, and probabilistic concepts such as independence and Markov processes are integrated into the text. Also, more attention than usual is paid to the role of algebras, and the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary.

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⭐The author was one of the great probability theorists of the century and this work was written near the end of his career.It is typical of his meticulous style, much in the spirit of pure mathematics.I have by no means finished reading this book, but I thought you might be interested to see its table of contents, which I have loosely transcribed using OCR(some of the symbols are not well rendered below)Introduction0. Conventions and Notation1 Notation: Euclidean space2. Operations involving ±infinity3.Inequalities and inclusions4.A space and its subsets5.Notation: generation of classes of sets6.Product sets7.Dot notation for an index set8.Notation: sets defined by conditions on functions9.Notation: open and closed sets10.Limit of a function at a point11.Metric spaces12.Standard metric space theorems13.Pseudometric spacesI. Operations on Sets1.Unions and intersections2.The symmetric difference operator Delta3.Limit operations on set sequences4.Probabilistic interpretation of sets and operations on themII. Classes of Subsets of a Space1.Set algebras2.Examples3.The generation of set algebras4.The Borel sets of a metric space5.Products of set algebras6.Monotone classes of setsIII. Set Functions1.Set function definitions2.Extension of a finitely additive set function3.Products of set functions4.Heuristics on sigma algebras and integration5.Measures and integrals on a countable space6.Independence and conditional probability (preliminary discussion)7.Dependence examples8.Inferior and superior limits of sequences of measurable sets9.Mathematical counterparts of coin tossing10.Setwise convergence of measure sequences11.Outer measure12.Outer measures of countable subsets of R13.Distance on a set algebra defined by a subadditive set function14.The pseudometric space defined by an outer measure15.Nonadditive set functionsIV. Measure Spaces 371.Completion of a measure space (S,S,Lambda)2.Generalization of length on R3.A general extension problem4.Extension of a measure defined on a set algebra5.Application to Borel measures6.Strengthening of Theorem 5 when the metric space S is complete and separable7.Continuity properties of monotone functions8.The correspondence between monotone increasing functions on Rand measures on B(R)9.Discrete and continuous distributions on R10.Lebesgue-Stieltjes measures on RN and their corresponding monotone functions11.Product measures12.Examples of measures on RN13.Marginal measures14.Coin tossing15.The Caratheodory measurability criterion16.Measure hullsV. Measurable Functions1.Function measurability2.Function measurability properties3.Measurability and sequential convergence4.Baire functions5.Joint distributions6.Measures on function (coordinate) space7.Applications of coordinate space measures8.Mutually independent random variables on a probability space9.Application of independence: the 0-1 law10.Applications of the 0-1 law11.A pseudometric for real valued measurable functions on a measurespace12.Convergence in measure13.Convergence in measure vs. almost everywhere convergence14.Almost everywhere convergence vs. uniform convergence15.Function measurability vs. continuity16.Measurable functions as approximated by continuous functions17.Essential supremum and infimum of a measurable function18.Essential supremum and infimum of a collection of measurable functionsVI. Integration1.The integral of a positive step function on a measure space (S,S,Lambda)2.The integral of a positive function3.Integration to the limit for monotone increasing sequencesof positive functions4.Final definition of the integral5.An elementary application of integration6.Set functions defined by integrals7.Uniform integrability test functions8.Integration to the limit for positive integrands9.The dominated convergence theorem10.Integration over product measures11.Jensen’s inequality12.Conjugate spaces and Holder’s inequality13.Minkowski’s inequality14.The L^p spaces as normed linear spaces15.Approximation of LP functions 9116.Uniform integrability17.Uniform integrability in terms of uniform integrability test functions18.L^1 convergence and uniform integrability19.The coordinate space context20.The Riemann integral21.Measure theory vs. premeasure theory analysisVII. Hilbert Space1.Analysis of L22.Hilbert space3.The distance from a subspace4.Projections5.Bounded linear functionals on lb6.Fourier series7.Fourier series properties8.Orthogonalization (Erhardt Schmidt procedure)9.Fourier trigonometric series10.Two trigonometric integrals11.Heuristic approach to the Fourier transform via Fourier series12.The Fourier-Plancherel theorem13.Ergodic theoremsVIII. Convergence of Measure Sequences1.Definition of convergence of a measure sequence2.Linear functionals on subsets of C(S)3.Generation of positive linear functionals by measures(S compact metric).4.C(S) convergence of sequences in M(S) (S compact metric)5.Metrization of M(S) to match C(S) convergence; compactness of Mc(S) (S compact metric)6.Properties of the function µ->µ[f ], from M(S), in the dm metric into R (S compact metric)7.Generation of positive linear functionals on C_o(S) by measures (S a locally compact but not compact separable metric space)8.C_o(S) and C_oo(S) convergence of sequences in M(S) (S a locallycompact but not compact separable metric space)9.Metrization of M(S) to match C_o(S) convergence; compactness of M_c(S) (S a locally compact but not compact separable metricspace, c a strictly positive number)10.Properties of the function µ->µ[f], from M(S) in the d_oM metricinto R (S a locally compact but not compact separable metric space)11.Stable C_o(S) convergence of sequences in M(S) (S a locally compact but not compact separable metric space)12.Metrization of M(S) to match stable C_o(S) convergence (S a locally compact but not compact separable metric space)13.Properties of the function from M(S) in the dm’ metric into R (S a locally compact but not compact separable metric space)14. Application to analytic and harmonic functionsIX. Signed Measures1.Range of values of a signed measure2.Positive and negative components of a signed measure3.Lattice property of the class of signed measures4.Absolute continuity and singularity of a signed measure5.Decomposition of a signed measure relative to a measure6.A basic preparatory result on singularity7.Integral representation of an absolutely continuous measure8.Bounded linear functionals on L’9.Sequences of signed measures10.Vitali-Hahn-Saks theorem (continued)11.Theorem 10 for signed measuresX. Measures and Functions of Bounded Variation on R1.Introduction2.Covering lemma3.Vitali covering of a set4.Derivation of Lebesgue-Stieltjes measures and of monotone functions5.Functions of bounded variation6.Functions of bounded variation vs. signed measures7.Absolute continuity and singularity of a function of bounded variation8.The convergence set of a sequence of monotone functions9.Helly’s compactness theorem for sequences of monotone functions10.Intervals of uniform convergence of a convergent sequence of monotone functions11.C(I) convergence of measure sequences on a compact interval I12.C_o(R) convergence of a measure sequence13.Stable C_0(R) convergence of a measure sequence14.The characteristic function of a measure15.Stable C_o(R) convergence of a sequence of probability distributions16.Application to a stable C_o(R) metrization of M(R)17.General approach to derivation18.A ratio limit lemma19.Application to the boundary limits of harmonic functionsXI. Conditional Expectations ; Martingale Theory1.Stochastic processes2.Conditional probability and expectation3 Conditional expectation properties4.Filtrations and adapted families of functions5.Martingale theory definitions6.Martingale examples7.Elementary properties of (sub- super-) martingales8.Optional times9.Optional time properties10.The optional sampling theorem11.The maximal submartingale inequality12.Upcrossings and convergence13.The submartingale upcrossing inequality14.Forward (sub- super-) martingale convergence15.Backward martingale convergence16.Backward supermartingale convergence17.Application of martingale theory to derivation18.Application of martingale theory to the 0-1 law19.Application of martingale theory to the strong law of large numbers20.Application of martingale theory to the convergence of infinite series21.Application of martingale theory to the boundary limits of harmonic functionsNotationIndex

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Free Download Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition in PDF format
Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition PDF Free Download
Download Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition 1994 PDF Free
Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition 1994 PDF Free Download
Download Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition PDF
Free Download Ebook Measure Theory (Graduate Texts in Mathematics, 143) 1994th Edition