Ebook: Measure, Integral and Probability

The key concept is that of measure which is first developed on the real line and then presented abstractly to provide an introduction to the foundations of probability theory (the Kolmogorov axioms) which in turn opens a route to many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities. Throughout, the development of the Lebesgue Integral provides the essential ideas: the role of basic convergence theorems, a discussion of modes of convergence for measurable functions, relations to the Riemann integral and the fundamental theorem of calculus, leading to the definition of Lebesgue spaces, the Fubini and Radon-Nikodym Theorems and their roles in describing the properties of random variables and their distributions. Applications to probability include laws of large numbers and the central limit theorem.

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The key concept is that of measure which is first developed on the real line and then presented abstractly to provide an introduction to the foundations of probability theory (the Kolmogorov axioms) which in turn opens a route to many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities. Throughout, the development of the Lebesgue Integral provides the essential ideas: the role of basic convergence theorems, a discussion of modes of convergence for measurable functions, relations to the Riemann integral and the fundamental theorem of calculus, leading to the definition of Lebesgue spaces, the Fubini and Radon-Nikodym Theorems and their roles in describing the properties of random variables and their distributions. Applications to probability include laws of large numbers and the central limit theorem.
Content:
Front Matter....Pages i-xi
Motivation and preliminaries....Pages 1-13
Measure....Pages 15-51
Measurable functions....Pages 53-69
Integral....Pages 71-114
Spaces of integrable functions....Pages 115-143
Product measures....Pages 145-169
Limit theorems....Pages 171-208
Solutions to exercises....Pages 209-217
Appendix....Pages 219-221
Back Matter....Pages 223-227

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The key concept is that of measure which is first developed on the real line and then presented abstractly to provide an introduction to the foundations of probability theory (the Kolmogorov axioms) which in turn opens a route to many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities. Throughout, the development of the Lebesgue Integral provides the essential ideas: the role of basic convergence theorems, a discussion of modes of convergence for measurable functions, relations to the Riemann integral and the fundamental theorem of calculus, leading to the definition of Lebesgue spaces, the Fubini and Radon-Nikodym Theorems and their roles in describing the properties of random variables and their distributions. Applications to probability include laws of large numbers and the central limit theorem.
Content:
Front Matter....Pages i-xi
Motivation and preliminaries....Pages 1-13
Measure....Pages 15-51
Measurable functions....Pages 53-69
Integral....Pages 71-114
Spaces of integrable functions....Pages 115-143
Product measures....Pages 145-169
Limit theorems....Pages 171-208
Solutions to exercises....Pages 209-217
Appendix....Pages 219-221
Back Matter....Pages 223-227
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