μ-Adequate Family of Measures

19.1 For Radon measures, the notion of induced measure is quite natural. If X is a locally compact Hausdorff space and Y a locally compact subspace, any Radon measure on X induces a Radon measure on y as follows. If f is continuous on Y with compact support, define g by g(x) = f(x) when x ∈ Y and g(x) = 0 otherwise. Then f ↦ ∫gdμ, is the measure induced by μ on Y.19.2 μ-dense families of compact sets, introduced in this section, will be needed later in the text.19.3 Given a positive Radon measure μ and a μ-dense class D of compact sets, there is a summable family (μ_α)α∈A of measures such that $$\mu = \sum\nolimits_{\alpha \in A} {{\mu_\alpha}}$$ that supp(μ_α) belongs to D, and that the sets supp(μ_α) form a locally countable class (Theorem 19.3.1).19.4 In this section, we study integration with respect to ∫λ _t d_μ(t), where t↦λ _t At is a μ-adequate family of measures. We prove a result which is similar to Fubini’s theorem (Theorem 19.4.2).19.5 We specialize the results of Section 19.4 to μ-adapted pairs.