Images of Measures

Summary

11.1 Given a measure μ on S in Ω and a mapping π from Ω into Ω′, we can try to define a measure on a semiring S′ in Ω′ by μ′(A) = μ(π⁻¹(A)). This will define a measure if certain conditions are satisfied, in which case we say that the pair (π, S′) is μ-suited. Then ∫ f dμ′ = ∫(f ○ π)dμ (Theorem 11.1.1).

11.2 In this section, we define compact classes, projective systems of measures, and prove Kolmogorov’s theorem, which gives a sufficient condition for a projective system of measures to define a measure called the projective limit of the system (Theorem 11.2.1). In particular, if for each i ∈ I μ _i is a positive measure with total mass 1 on S _i , and if μ _J = ⊗ _J μ _i for all finite subsets J ⊂ I, then {μ _J } has a projective limit (Theorem 11.2.2).

11.3 We apply the notion of image of a measure to Lebesgue measure on R.

11.4 This is a short introduction to ergodic theory. The main result of this section is Birkhoff’s ergodic theorem: If f: Ω → R is essentially μ-integrable and f _k = f ○ u^k (where u: Ω → Ω satisfies u(μ) = μ), then (1/n) $\sum\nolimits_{0 \leqslant k \leqslant n - 1} {{f_k}}$ converges to some essentially μ-integrable function f^* locally μ-a.e. and f^* = f^* ○ u locally μ-a.e.