232 Pages
    by CRC Press

    232 Pages
    by CRC Press

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    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 Measures
    σ-algebras as events
    Algebras, monotone classes, etc.
    Preliminaries on measures
    Outer measures and Caratheodory extension
    Lebesgue measure
    Bernoulli trials

    Measurable functions
    a.e. considerations

    Random Variables
    Distribution and expectation
    Independent events and tail σ-algebra
    Some distributions
    Conditional expectation

    Probability Measures on Product Spaces
    Product measures
    Joint distribution and independence
    Probability measures on infinite product spaces
    Kolmogorov consistency theorem

    Characteristics and Convergences
    Characteristic functions
    Modes of convergence
    Central limit theorem
    Law of large numbers

    Markov Chains
    Discrete time MC
    Classification of states
    Strong Markov property
    Stationary distribution
    Limit theorems

    Some Analysis
    Complex measures
    Lp spaces
    Radon–Nikodym theorem
    Change of variables
    The Riesz representation theorem

    Metric spaces
    Topological spaces
    The Stone–Weierstrass theorem





    Siva Athreya is a professor at the Indian Academy of Sciences, Bangalore, and V.S. Sunder is a professor at the Institute of Mathematical Sciences, Chennai.

    This textbook is suitable for a one-semester course on measure theory and probability for beginning graduate students in mathematics, probability and statistics. It can also be used as a textbook for advanced undergraduate students in mathematics … The topics are well selected to meet the needs of students who are interested in graduate studies in areas related to analysis, probability, stochastic processes and statistics … This makes the book student-friendly. A motivated student can use it by him- or herself to learn the topics well.

    —Yimin Xiao, Mathematical Reviews, 2010