Lebesgue differentiation theorem

If you average an integrable function over tinier and tinier balls around a point, the averages home in on the function's value there — at almost every point. This recovers the function from its integral the way the fundamental theorem of calculus recovers an integrand from its accumulated integral, but now for the rough, merely integrable functions Lebesgue theory admits, not only continuous ones.

Precisely, for f locally integrable on R^n, for almost every x the average of f over the ball B(x, r), namely (1 over the measure of the ball) times the integral of f over B(x, r), converges to f(x) as r goes to 0. The stronger Lebesgue point version asserts that the average of |f(y) - f(x)| over the ball also tends to 0 at almost every x — such x are called Lebesgue points.

This is the deep generalization of the fundamental theorem of calculus to the Lebesgue setting; for continuous f it is just continuity, but the theorem holds with no continuity at all. The phrase almost every is essential: it can fail on a measure-zero set, and indeed for the indicator of a set it fails exactly on the measure-theoretic boundary. The proof rests on the Hardy–Littlewood maximal inequality.