STAT 205 A - Problem Set 08
نویسنده
چکیده
this is true for indicator functions, since ∫ 1Adπ = π(A) = ∫ A hπ,μdμ by definition of the Radon-Nikodym derivative. Then by linearity of the integral, the claim is also true for simple functions. Then we can show the claim for non-negative measurable functions, by approximating by simple functions. Indeed, if f ≥ 0 is measurable, then there exists a sequence of simple functions 0 ≤ sn ↑ f such that ∫ sndπ ↑ ∫ fdπ. For all n, we have ∫ sndπ = ∫ snhπ,μ = ∫ snhπ,μdμ where 0 ≤ snhπ,μ ↑ fhπ,μ since hπ,μ ≥ 0. By the Monotone Convergence theorem, we have ∫ fdπ = ∫ fhπ,μdμ. Finally, to show the claim for any integrable function f , we can write f = f − f− where each term is non-negative, and apply the previous result to each term. Now we have for any integrable function f ∫ fdπ = ∫ fhπ,μdμ (1)
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