Consider the real-valued random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the function $f\colon\mathbb{R}\rightarrow [0,\infty)$. Let $P_{X,Y}$ be the joint probability distribution of $X,Y$. Suppose $E_{\mathbb{P}}[(Y-m)^2f(X)]:=\int_{\mathbb{R}^2}(y-m)^2f(x)dP_{X,Y}$ exists.
Consider the map $m\rightarrow E_{\mathbb{P}}[(Y-m)^2f(X)]$. I want to show that under some sufficient conditions this map is differentiable at each $m$. My questions is related to these sufficient conditions. The key part is the possibility of applying the Leibniz rule so that I can bring differentiation inside the integral.
Question:
(1) Is the wiki statement of the Leibniz rule for the Lebesgue integral correct? If not, can you provide a correct statement? I think, for example, that $\Omega$ is not a measure space but a sample space.
(2) How can I apply the statement to my case? In some books I have found as sufficient condition $E_{\mathbb{P}}(Y^2|X)$ bounded by a finite number but I don't understand why.