Differentiability of Functions Defined by Lebesgue Integrals

Question

Let $f:[0,1] \times \mathbb{R} \to \mathbb{R}$ be $C^1$ function and consider $F(y)= \int_0^1 f(x,y) dm(x)$. I want to show that $F$ is differentiable and $F'(y)= \int_0^1 \frac{\partial f}{\partial y}(x,y) dm(x)$.

I know that $$F'(y_0)=\lim_{y \to y_0} \frac{F(y)-F(y_0)}{y-y_0}=\lim_{y \to y_0} \int_0^1\frac{f(x,y)-f(x,y_0)}{y-y_0}$$

I'm having trouble throwing the limit into the integral, how can I justify this?

Answer 1

Answer in Leibniz rule for Lebesgue integrals