Lebesgue differentiation theorem

Let f be a locally integrable function on n with Lebesgue measureMathworldPlanetmath m, i.e. fLloc1(n). Lebesgue’s differentiationMathworldPlanetmath theorem basically says that for almost every x, the averagesMathworldPlanetmath


converge to 0 when Q is a cube containing x and m(Q)0.

Formally, this means that there is a set Nn with μ(N)=0, such that for every xN and ε>0, there exists δ>0 such that, for each cube Q with xQ and m(Q)<δ, we have


For n=1, this can be restated as an analogue of the fundamental theorem of calculus for Lebesgue integrals. Given a x0,


for almost every x.

Title Lebesgue differentiation theorem
Canonical name LebesgueDifferentiationTheorem
Date of creation 2013-03-22 13:27:36
Last modified on 2013-03-22 13:27:36
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Theorem
Classification msc 28A15