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

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

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