Lebesgue density theorem

Let $\mu$ be the Lebesgue measure on ${ℝ}^n$, and for a measurable set $A\subset {ℝ}^n$ define the density of $A$ in $ϵ$-neighborhood of $x\in {ℝ}^n$ by

$$d_{ϵ}(x)=\frac{\mu (A\cap B_{ϵ}(x))}{\mu (B_{ϵ}(x))}$$

where $B_{ϵ}(x)$ denotes the ball of radius $ϵ$ centered at $x$.

The Lebesgue density theorem asserts that for almost every point of $A$ the density

$$d(x)=\underset{ϵ\to 0}{lim}{d}_{ϵ}(x)$$

exists and is equal to $1$.

In other words, for every measurable set $A$ the density of $A$ is $0$ or $1$ almost everywhere. However, it is a curious fact that if $\mu (A)>0$ and $\mu ({ℝ}^n\setminus A)>0$, then there are always points of ${ℝ}^n$ where the density is neither $0$ nor $1$ [1, Lemma 4].

Title	Lebesgue density theorem
Canonical name	LebesgueDensityTheorem
Date of creation	2013-03-22 13:21:02
Last modified on	2013-03-22 13:21:02
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	7
Author	bbukh (348)
Entry type	Theorem
Classification	msc 28A75