Lebesgue point

From Wikipedia, the free encyclopedia

You have new messages (last change).

In mathematics, given a Lebesgue integrable function f, a point x in the domain of f is a Lebesgue point if

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

Here, B(x,r) is the ball centered at x with radius r, and | B(x,r) | is the Lebesgue measure of that ball. The Lebesgue points of f are thus points where f does not oscillate too much, in an average sense.

It can be shown that, given any f\in L^1(R^k), almost every x is a Lebesgue point.

Retrieved from "http://en.wikipedia.org../../../l/e/b/Lebesgue_point.html"
In other languages