Lebesgue point

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

[math]\displaystyle{ \lim_{r\rightarrow 0^+}\frac{1}{\lambda (B(x,r))}\int_{B(x,r)} \!|f(y)-f(x)|\,\mathrm{d}y=0. }[/math]

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

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