Prove continuity of averaging function for integrable $f$

Question

I want to prove the following statement which is part of a lemma in my textbook:

Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le r} |f(x-y)-f(x)| \, dy$$ for $r>0$

Show $M(r)$ is a continuous function for $r>0$

---

My try:

By changing of variable,

$$M(r)=\frac{1}{r^d}\int_{|y|\le r} |f(x-y)-f(x)| \, dy=\int_{|z|\le 1} |f(x+rz)-f(x)| \, dz$$

Hence fixing $r_1>0$, I get $$ |M(r_2)-M(r_1)|\le \int_{|z|\le 1} |f(x+r_2z)-f(x+r_1z)| \, dz $$

If $f$ is continuous, let $r_2$ closed enough to $r_1$, I get uniform continuous then the conclusion follows.

For integrable $f$ , I think I need to use Lusin theorem to approximate $f$ be a continuous function and use absolute continuity of $f$, which follows from the inegrability of $f$. But I am not sure how to argue it.

Answer 1

Hints: Set

$$M(r,f) := \frac{1}{r^d} \int_{|y| \leq r} |f(x-y)-f(x)| \, dy.$$

1. A very similar calculation as the one in the OP shows that $$|M(r,f)-M(r,g)| \leq \|f-g\|_{L^1}$$ for any two integrable functions $f,g$ and $r>0$.
2. Recall that the continuous compactly supported functions $C_c(\mathbb{R}^n)$ are dense in $L^1$. Choose a sequence $(f_n)_{n \in \mathbb{N}}$ such that $\|f_n-f\|_{L^1} \to 0$. Conclude from the first step and the fact that $$r \mapsto M(r,f_n)$$ is continuous for each $n \in \mathbb{N}$, that $$r \mapsto M(r,f)$$ is continuous.