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

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

Here, $B(x,r)$ is a ball centered at $x$ with radius $r>0$, and $|B(x,r)|$ 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 $f$.