Lebesgue point

In mathematics, given a locally Lebesgue integrable function $f$ on ${\mathbb{ℝ}}^k$, a point $x$ in the domain of $f$ is a Lebesgue point if

$\underset{r\to 0^{+}}{\lim }\frac{1}{\lambda (B(x,r))}\underset{B(x,r)}{\int }|f(y)-f(x)|\mathrm{d}y=0.$

Here, $B(x,r)$ is a ball centered at $x$ with radius $r>0$, and $\lambda (B(x,r))$ is its Lebesgue measure. The Lebesgue points of $f$ are thus points where $f$ does not oscillate too much, in an average sense.

The Lebesgue differentiation theorem states that, given any $f\in L^1({\mathbb{ℝ}}^k)$, almost every $x$ is a Lebesgue point of $f$.