Lebesgue Point

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. It can be shown that, given any f as above, almost every x is a Lebesgue point.

 

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