Lebesgue point

In mathematics, given a locally Lebesgue integrable function <math>f</math> on <math>\mathbb{R}^k</math>, a point <math>x</math> in the domain of <math>f</math> is a Lebesgue point if<ref>Template:Citation.</ref>

<math>\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>B(x,r)</math> is a ball centered at <math>x</math> with radius <math>r > 0</math>, and <math>\lambda (B(x,r))</math> is its Lebesgue measure. The Lebesgue points of <math>f</math> are thus points where <math>f</math> does not oscillate too much, in an average sense.<ref>Template:Citation.</ref>

The Lebesgue differentiation theorem states that, given any <math>f\in L^1(\mathbb{R}^k)</math>, almost every <math>x</math> is a Lebesgue point of <math>f</math>.<ref>Template:Citation.</ref>