Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $f(x, \cdot)$ is Borel-measurable for each $x$, and $f(\cdot, y)$ is continuous for every $y$.
Define $f_n: \mathbb{R}^2 \to \mathbb{R}$ by $f_n(x,y) = f\big(\frac{\lfloor nx \rfloor}{n}, y \big).$ I want to show that $f_n$ is Borel-measurable, but I'm not sure how to do this. Can anyone help?
$f_n(x,y)=\sum_k I_{[\frac k n, \frac {k+1} n)}(x) f(\frac k n , y)$. If you multiply a measurable function of $x$ by a measurable function of $y$ you get a measurable function of $(x,y)$. Can you complete the argument?