Prove $A(r)=\frac{1}{r^d} \int_{B(0,r)} |f(x-y)-f(y)|dy=0$ as $r \rightarrow 0$, f here is in $L^1(R^d)$ and x is a Lebesgue point for f.
I tried insert f(x) and use triangular inequality, but it fails as $A(r)=\frac{1}{r^d} \int_{B(0,r)} |f(x)-f(y)|dy$ as $r \rightarrow 0$ does not go to 0. Then I considered $$ f= \left\{ \begin{aligned} & 1, -1/2<x<1/2\\ & 0, o/w \end{aligned} \right. $$ x=2 is a Lebesgue point for f, define $A(r)=\frac{1}{r^d} \int_{B(0,r)} |f(2-y)-f(y)|dy=0$ which seems to be a conterexample.
Where did I go wrong? And how should I prove it? Any help is appreciated.