Appendix B

Measure and integration

The axiomatic approach to probability by Andrey Kolmogorov (1903–1987) makes essential use of the measure theory. In this appendix we review the aspects of the theory that are relevant to us. We do not prove everything and refer the interested reader for proofs and further study to one of the many volumes on this now classic subject, see e.g. [7, 27].

B.1   Measures

B.1.1   Basic properties

Here Ω shall denote a generic set. For a generic subset E of Ω, Ec := Ω \ E denotes the complement of E in Ω and Images(Ω) denotes the family of all subsets of Ω. A family Images of subsets of Ω is then a subset of Images(Ω), ImagesImages(Ω). We say that a family ImagesImages(Ω) of subsets of a set Ω is an algebra if , Ω and EF, EF and Ec whenever E, F .

Definition B.1 We say that is a σ-algebra ...

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