About Lebesgue points of a function and a subset

Question

Let $f$ be a non-negative function on $\mathbb{R}^d$ with$\int_{\mathbb{R}^d}f(x)dx =1$.
Let $x_0\in\mathbb{R}^d$ be a Lebesgue point of $f$, i.e. we have $ \lim_{r\to 0}\frac{1}{m(B(r,x_0))}\int_{B(r,x_0)}\lvert f(x)-f(x_0)\rvert dx =0 $ where $m$ is the $d$-dimentional Lebesgue measure.
My question is:
Can we prove that $x_0$ is a Lebesgue point of $\{f>0\}$?

Answer 1

Assuming a Lebesgue point for the set $E$ is a Lebesgue point for $\chi_E$:

No, in general. Say $f\in C_c(\Bbb R)$. Then every point is a Lebesgue point for $f$, but not every point is a Lebesgue point for $\{f>0\}$.

Yes if you assume $f(x_0)>0$: Let $E=\{f>0\}$. Then $$\frac1{|B|}\int_B|f(x_0)-f|\ge f(x_0)\frac{|B\setminus E|}{|B|},$$so $|B\setminus E|/|B|\to0$ as $B\to x_0$, hence $|B\cap E|/|B|\to1$.