# Reference for "continuity in mean" of integrable functions

In the measury theory class I'm taking we proved the following theorem

Let $$f \in L^1(\mathbb{R^n})$$. Then for almost every $$x \in \mathbb{R^n}$$ it holds that $$\lim_{r\downarrow 0} \frac{1}{V_r}\int_{B_r(x)}dy \ |f(y)-f(x)| = 0 ,$$ where $$V_r > 0$$ is the Lebesgue measure of $$B_r(x)$$.

From this one obtains $$f(x)=\lim_{r\downarrow 0}\frac{1}{V_r}\int_{B_r(x)}dy \ f(y) \ a.e.$$

This is referred to as "continuity in mean" of integrable functions. We used this to prove the fundamental theorem of calculus analogue for the Lebesgue integral or to obtain the fundamental lemma of variational calculus. It seems like to be an important property.

I didn't find this result in other texts on the topic and googling "continuity in mean" doesn't yield any other source referencing this. Is there a different name of the property and are there other sources where this result is used?

• The points where the first limit exists are called Lebesgue points and the statement that a.e. point is a Lebesgue point is known as Lebesgue differentiation theorem. Jan 28 '19 at 15:15