Lebesgue point

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^[1]

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

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.^[2]

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 ${\displaystyle f}$.^[3]