A First Course in Probability and Markov Chains

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, E^c := Ω \ E denotes the complement of E in Ω and (Ω) denotes the family of all subsets of Ω. A family of subsets of Ω is then a subset of (Ω), ⊂ (Ω). We say that a family ⊂ (Ω) of subsets of a set Ω is an algebra if , Ω and E ∪ F, E ∩ F and E^c whenever E, F .

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