Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions:
- for each $x$, $f(x, y)$ is an integrable function of $y$.
- $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded function of $(x, y)$.
Then show that $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a measurable unction of $y$ for each $x$.
I think the first condition is $$\int_{[0, 1]\times[0, 1]} f(x, y) ~ dy \quad \text{ exists and finite.}$$
Moreover, the second condition is, for a constant $M\in\mathbb{R}^1$, $$\left|\frac{\partial{}f(x, y)}{\partial{}x}\right|\le{}M$$
My changing of conditions are correct?
And $$\frac{\partial f(x,y)}{\partial x}=\lim_{h\to0}\frac{f(x+h,y)-f(x,y)}h. $$ is why measurable?