Lebesgue point

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

Here, B ( x , r ) is a ball centered at x with radius r > 0 , and | 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 ∈ L 1 ( R k ) , almost every x is a Lebesgue point of f .