Fatou’s Lemma and Lebesgue’s convergence theorem for measures

Measure Theory – Part 11 – Proof of Lebesgue’s Dominated Convergence Theorem
Measure Theory – Part 11 – Proof of Lebesgue’s Dominated Convergence Theorem

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2000, Journal of Applied Mathematics and Stochastic Analysis

Analogues of Fatou's Lemma and Lebesgue's convergence theorems are established for∫fdμnwhen{μn}is a sequence of measures. A generalized Dominated Convergence Theorem is also proved for the asymptotic behavior of∫fndμnand the latter is shown to be a special case of a more general result established in vector lattices and related to the Dunford-Pettis property in Banach spaces.

Mathematics of Operations Research

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We consider a sequence{μn}of (nonnegative) measures on a general measurable space(X,ℬ). We establish sufficient conditions for setwise convergence and convergence in total variation.

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Abstract. This paper studies the expected total cost (ETC) criterion for discrete-time Markov control processes on Borel spaces, and possibly unbounded cost-per-stage functions. It presents optimality results which include conditions for a control policy to be ETC-optimal and for the …

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