Differentiation under (measure theoretical) integral sign

probability-theory reference-request soft-question measure-theory

Solution 1:

I found the answer to your question on p.142 of "Probability Theory: A Comprehensive Course, Ed. 2" by Klenke. Its called the Differentiation Lemma in that text. You've stated, almost word for word, the pre-conditions of that lemma. Here is a link that describes the conditions on exchanging integration and diffeentiation.

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