Suppose $f$ and $g$ are in $L^1(\mathbb{R}^n,m)$ where $m$ is a Lebesgue measure. And suppose that $f=g$ almost everywhere with respect to $m$.
Let $L_f$ and $L_g$ be the Lebesgue sets for f and g in $L^1 (loc) $ respectively. My question is :
Can $L_f$ be described in term of $L_g$? I tried to use the formula for the average value of f : $(A_rf)(x)=\frac{1}{m(B(r,x))}\int_{B(r,x)}f(y)\;dm(y)$
Thanks in advance
Yes you can describe $L_f $ in terms of $L_g$ . Since $f=g$ almost everywhere , let
$A=\{x \in \Bbb R^n : f(x)=g(x)\}$ then $m(\Bbb R^n \backslash A)=0.$ So
$(A_{r}f)(x)=\frac{1}{m(B(r,x))}\int_{B(r,x))} f(y)dm(y)=\frac{1}{m(B(r,x))}\int_{B(r,x)\cap A} f(y)dm(y) +$
$\frac{1}{m(B(r,x))} \int_{B(r,x)\cap \Bbb R^n\backslash A} f(y)dm(y) = \frac{1}{m(B(r,x))}\int_{B(r,x)\cap A} f(y)dm(y)$ =$\frac{1}{m(B(r,x))}\int_{B(r,x)\cap A} g(y)dm(y)$
=$\frac{1}{m(B(r,x))}\int_{B(r,x)} g(y)dm(y)=(A_rg)(x)$.