my question is as follows. 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 I do not understand what to do

Thanks

My question is as follows. Consider an $L^\infty$ function $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ such that, for almost all $y$, the function $f({\cdot}, y)$ is continuous.

$\DeclareMathOperator\conv{conv}$Let $\conv D$ 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 $$\bigcap_{r,r'>0}\bigcap_N\conv f(B_r(x_0),B_{r'}(y_0)\setminus N)=\bigcap_{r'>0}\bigcap_N\conv f(x_0,B_{r'}(y_0)\backslash N).$$ Here $\bigcap_N$ stands for the intersection over all measure-$0$ 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 I do not understand what to do.