Lebesgue point

In mathematics, given a Lebesgue integrable function $f$, a point $x$ in the domain of $f$ is a Lebesgue point if

:$lim\_\{r\; ightarrow\; 0^+\}frac\{1\}int\_\{B(x,r)\}\; !|f(y)-f(x)|,dy=0.$

Here, $B(x,r)$ is the ball centered at $x$ with radius $r$, and $|B(x,r)|$ is the Lebesgue measure of that ball. The Lebesgue points of $f$ are thus points where $f$ does not oscillate too much, in an average sense.

It can be shown that, given any $fin\; L^1(R^k)$, almost every $x$ is a Lebesgue point.