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
 * $$\lim_{r\rightarrow 0^+}\frac{1}{\lambda (B(x,r))}\int_{B(x,r)} \!|f(y)-f(x)|\,\mathrm{d}y=0.$$

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