Образ измеримой функции

[pogulyat_vyshel]

Consider a measurable function $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n,\quad f=f(x,y).$

1) For almost all $(x,y)$ we have $|f(x,y)|\le const$;

2) for almost all $y$ the function $f$ is a continuous function of $x$.

Let $\mathrm{conv}\,D$ stand for the closed convex hull of a set $D$ and let $B_r(x)\subset\mathbb{R}^n$ be the open ball
with radius $r>0$ and with center at $x$.

I want to write

$$\bigcap_{r,r'>0}\bigcap_N\mathrm{conv}\,f(B_r(x_0),B_{r'}(y_0)\backslash N)=\bigcap_{r'>0}\bigcap_N\mathrm{conv}\,f(x_0,B_{r'}(y_0)\backslash N).$$

Here $\bigcap_N$ stands for the intersection over all measure null sets $N\subset\mathbb{R}^n$.
I feel that there must be some uniform continuity conditions on $f$ for this equality to be valid but can not prove anything

[pogulyat_vyshel]

сделано, можно закрывать