Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
5 - Absolute continuity and related topics
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
Summary
Signed and complex measures
Relaxation of the requirement of a measure that it be nonnegative yields what is usually called a signed measure. Specifically this is an extended real-valued, countably additive set function μ on a class ε (containing ∅), such that μ(∅) = 0, and such that μ assumes at most one of the values +∞ and −∞ on ε. As for measures, a signed measure μ defined on a class ε, is called finite on ε if |μ(E)| < ∞, for each E ∈ ε, and σ-finite if for each E ∈ ε there is a sequence {En}∞n=1 of sets in ε with E ⊂ ∪∞n=1En and |μ(En)| < ∞, that is, if E can be covered by the union of a sequence of sets with finite (signed) measure. It will usually be assumed that the class on which μ is defined is a σ-ring or σ-field.
Some of the important properties of measures (see Section 2.2) hold also for signed measures. In particular a signed measure is subtractive and continuous from below and above. The basic properties of signed measures are given in the following theorem.
Theorem 5.1.1Let μ be a signed measure on a σ-ring S.
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- A Basic Course in Measure and ProbabilityTheory for Applications, pp. 86 - 117Publisher: Cambridge University PressPrint publication year: 2014