I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce a reference for the measure theoretic version of it?
The statement on wiki is as follows:
Let $X$ be an open subset of $\mathbb{R}$ , and $\Omega$ be
a measure space. Suppose $f: X \times \Omega \rightarrow \mathbb{R} $ satisfies the following conditions:
::(1) $f(x,\omega)$ is a Lebesgue-integrable function of $\omega$ for each $x \in X$
::(2) For almost all $\omega \in \Omega$ , the derivative $f_x$ exists for all $x \in X$
::(3) There is an integrable function $ \theta: \Omega \rightarrow \mathbb{R}$ such that $|f_x(x,\omega)| \leq \theta ( \omega)$ for all $x \in X$
Then for all $x \in X$
::$ \frac{\mathrm{d}}{\mathrm{d} x} \int_{\Omega} \, f(x, \omega) \mathrm{d} \omega = \int_{\Omega} \, f_x ( x, \omega) \mathrm{d} \omega $
Best Answer
check this link out, specially the proposition in section 2
http://people.hss.caltech.edu/~kcb/Notes/LeibnizRule.pdf