I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ is $\mathcal{M}_{[0,1]}$-measurable. Further suppose that $\frac{\partial f}{\partial y}$ exists and is bounded on $[0,1] \times (0,1)$.
Show that $$\frac{d}{dy}\int_{0}^{1}f(x,y) dx = \int_{0}^{1}\frac{\partial f}{\partial y}(x,y)dx$$.
I assume that the Lebesgue-$[0,1]$ notation means that it is Lebesgue measurable over $[0,1]$. I am pretty sure that I need to use the DCT, but how do I go about doing so with this? Do I just make some arbitrary function and claim that it bounds this rectangle?