Description
Taylor & Francis Ltd Measure And Probability 2009 Edition by Siva Athreya, V. S. Sunder
This book covers the fundamentals of measure theory and probability theory. It begins with the construction of Lebesgue measure via Caratheodory's outer measure approach and goes on to discuss integration and standard convergence theorems and contains an entire chapter devoted to complex measures, Lp spaces, Radon-Nikodym theorem, and the Riesz representation theorem. It presents the elements of probability theory, the law of large numbers, and central limit theorem. The book then discusses discrete time Markov chains, stationary distributions and limit theorems. The appendix covers many basic topics such as metric spaces, topological spaces and the Stone-Weierstrass theorem. Probabilities and MeasuresIntroduction-algebras as eventsAlgebras, monotone classes, etc.Preliminaries on measuresOuter measures and Caratheodory extensionLebesgue measureRegularityBernoulli trials IntegrationMeasurable functionsIntegrationa.e. considerationsRandom VariablesDistribution and expectationIndependent events and tail -algebraSome distributionsConditional expectationProbability Measures on Product SpacesProduct measuresJoint distribution and independenceProbability measures on infinite product spacesKolmogorov consistency theorem Characteristics and ConvergencesCharacteristic functionsModes of convergenceCentral limit theoremLaw of large numbersMarkov ChainsDiscrete time MCExamplesClassification of statesStrong Markov propertyStationary distributionLimit theoremsSome AnalysisComplex measuresLp spacesRadon-Nikodym theoremChange of variablesDifferentiationThe Riesz representation theoremAppendixMetric spacesTopological spacesCompactnessThe Stone-Weierstrass theoremTablesReferencesIndex