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?

  • 2
    $\begingroup$ 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. $\endgroup$
    – MaoWao
    Jan 28 '19 at 15:15

I think continuity in mean tends to get used in Probability texts only, and even then not that frequently, which might be why you've had trouble googling it. This is more commonly referred to as the average value of a function. In one dimension it's just the Mean Value Theorem.

See here: Why do we use integration to calculate the average of a function? for a nice explanation as to why the idea of using the integral to obtain the average comes about; it's easy to see from that how this generalises up to higher dimensions by integration over regions (typically balls, but you could use other shaped regions if you had a reason).

The average value of a function turns up a lot in Harmonic Analysis (and for harmonic functions you get the appealing result that the average can be calcuated by considering values just on the boundary of the ball) -- google "mean value harmonic function" for a lot of relevant results.


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