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A reprint of the original publication in Grundlehren der mathematischen Wissenschafen, Volume 262. Topics covered include green functions, the Dirichlet problem for relative harmonic functions, the fundamental convergence theorem and the reduction operation, and infirma of families of superharmonic functions. Softcover. ... Read more

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The theory of stochastic processes has developed so much in the last twenty years that the need for a systematic account of the subject has been felt, particularly by students and instructors of probability. This book fills that need. While even elementary definitions and theorems are stated in detail, this is not recommended as a first text in probability and there has been no compromise with the mathematics of probability. Since readers complained that omission of certain mathematical detail increased the obscurity of the subject, the text contains various mathematical points that might otherwise seem extraneous. A supplement includes a treatment of the various aspects of measure theory. A chapter on the specialized problem of prediction theory has also been included and references to the literature and historical remarks have been collected in the Appendix. ... Read more

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This text accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes and conditional expectations are integrated into the text rather than being relegated to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary. ... Read more

Customer Reviews (2)

Original presentation of Measure Theory
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)

Introduction

0. Conventions and Notation
1 Notation: Euclidean space
2. Operations involving ±infinity
3.Inequalities and inclusions
4.A space and its subsets
5.Notation: generation of classes of sets
6.Product sets
7.Dot notation for an index set
8.Notation: sets defined by conditions on functions
9.Notation: open and closed sets
10.Limit of a function at a point
11.Metric spaces
12.Standard metric space theorems
13.Pseudometric spaces

I. Operations on Sets
1.Unions and intersections
2.The symmetric difference operator Delta
3.Limit operations on set sequences
4.Probabilistic interpretation of sets and operations on them

II. Classes of Subsets of a Space
1.Set algebras
2.Examples
3.The generation of set algebras
4.The Borel sets of a metric space
5.Products of set algebras
6.Monotone classes of sets

III. Set Functions
1.Set function definitions
2.Extension of a finitely additive set function
3.Products of set functions
4.Heuristics on sigma algebras and integration
5.Measures and integrals on a countable space
6.Independence and conditional probability (preliminary discussion)
7.Dependence examples
8.Inferior and superior limits of sequences of measurable sets
9.Mathematical counterparts of coin tossing
10.Setwise convergence of measure sequences
11.Outer measure
12.Outer measures of countable subsets of R
13.Distance on a set algebra defined by a subadditive set function
14.The pseudometric space defined by an outer measure
15.Nonadditive set functions

IV. Measure Spaces 37
1.Completion of a measure space (S,S,Lambda)
2.Generalization of length on R
3.A general extension problem
4.Extension of a measure defined on a set algebra
5.Application to Borel measures
6.Strengthening of Theorem 5 when the metric space S is complete and separable
7.Continuity properties of monotone functions
8.The correspondence between monotone increasing functions on R
and measures on B(R)
9.Discrete and continuous distributions on R
10.Lebesgue-Stieltjes measures on RN and their corresponding monotone functions
11.Product measures
12.Examples of measures on RN
13.Marginal measures
14.Coin tossing
15.The Caratheodory measurability criterion
16.Measure hulls

V. Measurable Functions
1.Function measurability
2.Function measurability properties
3.Measurability and sequential convergence
4.Baire functions
5.Joint distributions
6.Measures on function (coordinate) space
7.Applications of coordinate space measures
8.Mutually independent random variables on a probability space
9.Application of independence: the 0-1 law
10.Applications of the 0-1 law
11.A pseudometric for real valued measurable functions on a measurespace
12.Convergence in measure
13.Convergence in measure vs. almost everywhere convergence
14.Almost everywhere convergence vs. uniform convergence
15.Function measurability vs. continuity
16.Measurable functions as approximated by continuous functions
17.Essential supremum and infimum of a measurable function
18.Essential supremum and infimum of a collection of measurable functions

VI. Integration
1.The integral of a positive step function on a measure space (S,S,Lambda)
2.The integral of a positive function
3.Integration to the limit for monotone increasing sequences
of positive functions
4.Final definition of the integral
5.An elementary application of integration
6.Set functions defined by integrals
7.Uniform integrability test functions
8.Integration to the limit for positive integrands
9.The dominated convergence theorem
10.Integration over product measures
11.Jensen's inequality
12.Conjugate spaces and Holder's inequality
13.Minkowski's inequality
14.The L^p spaces as normed linear spaces
15.Approximation of LP functions 91
16.Uniform integrability
17.Uniform integrability in terms of uniform integrability test functions
18.L^1 convergence and uniform integrability
19.The coordinate space context
20.The Riemann integral
21.Measure theory vs. premeasure theory analysis

VII. Hilbert Space
1.Analysis of L2
2.Hilbert space
3.The distance from a subspace
4.Projections
5.Bounded linear functionals on lb
6.Fourier series
7.Fourier series properties
8.Orthogonalization (Erhardt Schmidt procedure)
9.Fourier trigonometric series
10.Two trigonometric integrals
11.Heuristic approach to the Fourier transform via Fourier series
12.The Fourier-Plancherel theorem
13.Ergodic theorems

VIII. Convergence of Measure Sequences
1.Definition of convergence of a measure sequence
2.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 metric
space, c a strictly positive number)
10.Properties of the function µ->µ[f], from M(S) in the d_oM metric
into 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 functions

IX. Signed Measures
1.Range of values of a signed measure
2.Positive and negative components of a signed measure
3.Lattice property of the class of signed measures
4.Absolute continuity and singularity of a signed measure
5.Decomposition of a signed measure relative to a measure
6.A basic preparatory result on singularity
7.Integral representation of an absolutely continuous measure
8.Bounded linear functionals on L'
9.Sequences of signed measures
10.Vitali-Hahn-Saks theorem (continued)
11.Theorem 10 for signed measures

X. Measures and Functions of Bounded Variation on R
1.Introduction
2.Covering lemma
3.Vitali covering of a set
4.Derivation of Lebesgue-Stieltjes measures and of monotone functions
5.Functions of bounded variation
6.Functions of bounded variation vs. signed measures
7.Absolute continuity and singularity of a function of bounded variation
8.The convergence set of a sequence of monotone functions
9.Helly's compactness theorem for sequences of monotone functions
10.Intervals of uniform convergence of a convergent sequence of monotone functions
11.C(I) convergence of measure sequences on a compact interval I
12.C_o(R) convergence of a measure sequence
13.Stable C_0(R) convergence of a measure sequence
14.The characteristic function of a measure
15.Stable C_o(R) convergence of a sequence of probability distributions
16.Application to a stable C_o(R) metrization of M(R)
17.General approach to derivation
18.A ratio limit lemma
19.Application to the boundary limits of harmonic functions

XI. Conditional Expectations ; Martingale Theory
1.Stochastic processes
2.Conditional probability and expectation
3 Conditional expectation properties
4.Filtrations and adapted families of functions
5.Martingale theory definitions
6.Martingale examples
7.Elementary properties of (sub- super-) martingales
8.Optional times
9.Optional time properties
10.The optional sampling theorem
11.The maximal submartingale inequality
12.Upcrossings and convergence
13.The submartingale upcrossing inequality
14.Forward (sub- super-) martingale convergence
15.Backward martingale convergence
16.Backward supermartingale convergence
17.Application of martingale theory to derivation
18.Application of martingale theory to the 0-1 law
19.Application of martingale theory to the strong law of large numbers
20.Application of martingale theory to the convergence of infinite series
21.Application of martingale theory to the boundary limits of harmonic functions

Notation
Index

Measure Theory
This book will be useful to anyone who will be working in probabilityand is already acquainted with measure theory.It includes resultsthat are not standard material for a real analysis text but that areof interest inprobability. The author, who needsno introduction, uses simple examples(coin tossing, card shuffling)to illustrate how the "interesting,but imprecise real world" isrepresented in the "duller but moreprecise mathematical world", aswell as the importance of keepingthem apart.I know of no otherpublication where this is done aseffectively and as beautifully asit is presented here.Each proof inthis book is actually a sketchof its proof.This means that the readerdoes not get lost in detailsand sees the concepts transparently. Theauthor also does a good jobof limiting himself to what belongs to thetitle, as opposed todeveloping what would belong in a probability text.

I recommend this book to anyone interested in measure theory, whether or not their interest extends to probability. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Andrew Solomon, Joseph Leo Doob, William S. Dix, Wolfgang F. Danspeckgruber, Kaushik Basu, Carlos Fernández-Pello, Peter Schneider, Thomas J. Sargent, Yuli Tamir, Laurence Jonathan Cohen, Mia Bloom, Dean Rader, Giuliano Preparata, Cesare Segre, Josue Lajeunesse, Frank Gohlke, Clayne L. Pope, Merritt Bucholz, Stephen Scott, Fionnuala Ní Aoláin, Melissa Lane, Sylvia Lavin, Lawrence Sager, Ping-Hui Liao, F. T. Prince, James Love, Tomas Lindahl, Birgit Krawietz, Mogens Herman Hansen, Kim Jun-Yop. Excerpt:Andrew Solomon (born October 30, 1963) is a New York -born bisexual writer on politics, culture, and psychiatry who lives in New York and London . He has written for publications such as the New York Times , The New Yorker , and Artforum , on topics including depression , Soviet artists, the cultural rebirth of Afghanistan, Libyan politics, and Deaf politics. He is also a Contributing Writer for Travel and Leisure . In 2008, he was awarded the Humanitarian Award of the Society of Biological Psychiatry for his contributions to the field of mental health. He has a staff appointment as a Lecturer in Psychiatry at Cornell Medical School (Weill-Cornell Medical College).His most recent book, The Noonday Demon: An Atlas of Depression , won the 2001 National Book Award ; it was a finalist for the 2002 Pulitzer Prize , and has been published 24 languages.Education Solomon attended the private preparatory school Horace Mann , graduating cum laude[citation needed December 2009 . He received a Bachelor of Arts degree in English from Yale University in 1985, graduating magna cum laude. He studied English at Jesus College, Cambridge , earning the top first-class degree in English in his year, and an MA. As of 2009, he is pursuing a PhD at Jesus College, Cambridge in psychology, working on attac... ... Read more

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Chapters: Vilayanur S. Ramachandran, Léon Brillouin, Tzvetan Todorov, Joseph Leo Doob, Louis H. Galbreath, David H. French, Maynard Solomon, Charles Lemert, Billy Lauder, Srully Blotnick, Alberto Campo Baeza, Anatole le Braz, Veljko Rus, Yadin Dudai, Henri Cole, Reuven Rubinstein, Moshe Lewin, Geoffrey Hosking, Marcos Prado Troyjo, Dieter Henrich, Ping-Hui Liao, Michel Loève, Piotr Sztompka, James Kendall. Source: Wikipedia. Pages: 91. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Vilayanur Subramanian "Rama" Ramachandran is a neurologist best known for his work in the fields of behavioral neurology and psychophysics. He is currently the Director of the Center for Brain and Cognition, Professor in the Psychology Department and Neurosciences Program at the University of California, San Diego, and Adjunct Professor of Biology at the Salk Institute for Biological Studies. Ramachandran initially obtained an M.D. at Stanley Medical College in Madras, India, and subsequently obtained a Ph.D. from Trinity College at the University of Cambridge. Ramachandrans early work was on visual perception but he is best known for his experiments in behavioral neurology which, despite their apparent simplicity, have had a profound impact on the way we think about the brain. Ramachandran has been elected to fellowships at All Souls College, Oxford, and the Royal Institution, London (which also awarded him the Henry Dale Medal). He gave the 2003 BBC Reith Lectures and was conferred the title of Padma Bhushan by the President of India in 2007. He has been called The Marco Polo of neuroscience by Richard Dawkins and "the modern Paul Broca" by Eric Kandel. Newsweek magazine named him a member of "The Century Club", one of the "hundred most prominent people to watch" in the 21st century. V.S. Ramachandran was bor...More: http://booksllc.net/?id=417063 ... Read more