Measure, integral, and probability

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Measure, integral, and probability / Marek Capiński and Ekkehard Kopp

資料種別:

電子ブック

出版情報:

London : Springer, [20--]

形態:

1 online resource (xi, 227 p.)

シリーズ名:

Springer undergraduate mathematics series <BA34974319>

著者名:

ISBN:

9781447136316 [1447136314] (: electronic bk)

書誌ID:

TT00017804

子書誌情報

フルテキスト

所蔵情報

書誌詳細

注記:

Includes index
原資料の出版事項: London : Springer , c1999
Description based on online resource (viewed on December 4, 2019)

主題:

Integrals, Generalized; Measure theory; Probabilities

言語:

英語

目次情報:

Motivation and preliminaries / 1：

Notation and basic set theory / 1.1：

Sets and functions / 1.1.1：

Countable and uncountable sets in <$>\cal {R}<$> / 1.1.2：

Topological properties of sets in <$>\cal {R}<$> / 1.1.3：

The Riemann integral: scope and limitations / 1.2：

Choosing numbers at random / 1.3：

Measure / 2：

Null sets / 2.1：

Outer measure / 2.2：

Lebesgue measurable sets and Lebesgue measure / 2.3：

Basic properties of Lebesgue measure / 2.4：

Borel sets / 2.5：

Probability / 2.6：

Probability space / 2.6.1：

Events: conditioning and independence / 2.6.2：

Proofs of propositions / 2.7：

Measurable functions / 3：

The extended real line / 3.1：

Definition / 3.2：

Examples / 3.3：

Properties / 3.4：

Random variables / 3.5：

Sigma fields generated by random variables / 3.5.2：

Probability distributions / 3.5.3：

Independence of random variables / 3.5.4：

Integral / 3.6：

Definition of the integral / 4.1：

Monotone Convergence Theorems / 4.2：

Integrable functions / 4.3：

The Dominated Convergence Theorem / 4.4：

Relation to the Riemann integral / 4.5：

Approximation of measurable functions / 4.6：

Integration with respect to probability distributions / 4.7：

Absolutely continuous measures: examples of densities / 4.7.2：

Expectation of a random variable / 4.7.3：

Characteristic function / 4.7.4：

Spaces of integrable functions / 4.8：

The space L¹ / 5.1：

The Hilbert space L² / 5.2：

Properties of the L²-norm / 5.2.1：

Inner product spaces / 5.2.2：

Orthogonality / 5.2.3：

The L^p spaces: completeness / 5.3：

Moments / 5.4：

Independence / 5.4.2：

Product measures / 5.5：

Multi-dimensional Lebesgue measure / 6.1：

Product σ-fields / 6.2：

Construction of the product measure / 6.3：

Fubini's Theorem / 6.4：

Joint distributions / 6.5：

Independence again / 6.5.2：

Conditional probability / 6.5.3：

Characteristic functions determine distributions / 6.5.4：

Limit theorems / 6.6：

Modes of convergence / 7.1：

Convergence in probability / 7.2：

Weak law of large numbers / 7.2.2：

Borel-Cantelli lemmas / 7.2.3：

Strong law of large numbers / 7.2.4：

Weak convergence / 7.2.5：

Central Limit Theorem / 7.2.6：

Solutions to exercises / 7.3：

Appendix / 9：

References

Index

Motivation and preliminaries / 1：

Notation and basic set theory / 1.1：

Sets and functions / 1.1.1：

Countable and uncountable sets in <$>\cal {R}<$> / 1.1.2：

Topological properties of sets in <$>\cal {R}<$> / 1.1.3：

The Riemann integral: scope and limitations / 1.2：

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