Measures and Probabilities pp 175-192 | Cite as

# Induced Measures and Product Measures

## 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′′) = *V*µ′ ⊗ *V*µ′′ (Theorem 9.2.1). If *f*: Ω ↦ [0, +∞] is µ-measurable and µ-moderate, then ∫^{*} *f* *dV*µ = ∫^{*} *dV*µ′(*x*′) ∫^{*} *f*(*x*′,*x*′′) *dV*µ′′(*x*′′) can be computed by means of iterated integrals (Theorem 9.2.5). Likewise, we can compute ∫ *f* *d*µ 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

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