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Induced Measures and Product Measures

  • Michel Simonnet
Part of the Universitext book series (UTX)

Summary

9.1 Let µ be a measure on a semiring S whose underlying set is X. Let Y be a µ-measurable subset of X. Let T be a semiring whose underlying set is Y. Assume that µ, Y, and T are endowed with adequate properties. We define the measure µ/ T induced by µ on T and see how to integrate with respect to this induced measure (Theorem 9.1.1).

9.2 Let µ′, µ′′ be two measures on semirings S′, S′′ (with Ω′, Ω′′ as their underlying sets, respectively). The measure µ: A′ × A′′ ↦ µ′(A′) µ′′ (A′′) on the semiring S = {A′ × A′′: A′ ∈ S′, A′′ ∈ S′′} (whose underlying set is Ω = Ω′ × Ω′′) is called the product of µ′ and µ′′.V(µ′ ⊗ µ′′) = Vµ′ ⊗ Vµ′′ (Theorem 9.2.1). If f: Ω ↦ [0, +∞] is µ-measurable and µ-moderate, then ∫* f dVµ = ∫* dVµ′(x′) ∫* f(x′,x′′) dVµ′′(x′′) can be computed by means of iterated integrals (Theorem 9.2.5). Likewise, we can compute ∫ f dµ for every µ-integrable mapping from Ω into a Banach space (Theorem 9.2.4, Fubini’s).

9.3 We define Lebesgue measure on an open subset Ω of R k (with k ≥ 1).

Keywords

Banach Space Lebesgue Measure Product Measure Metrizable Space Countable Union 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Michel Simonnet
    • 1
  1. 1.Department of MathematicsUniversity of DakarSenegal

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