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 ○ uk (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.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). Images of Measures. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_11
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94644-3
Online ISBN: 978-1-4612-4012-9
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