Lebesgue Measure - Main Theorem

TheoremLebesgue Density Theorem

Let $E \subseteq \mathbb{R}$ be a Lebesgue measurable set. For $x \in \mathbb{R}$ , define the density of $E$ at $x$ as: $d(E, x) = \lim_{h \to 0} \frac{\lambda(E \cap (x-h, x+h))}{2h}$ when this limit exists.

Then for almost every $x \in E$ , we have $d(E, x) = 1$ , and for almost every $x \in E^c$ , we have $d(E, x) = 0$ .

This fundamental result characterizes measurable sets in terms of their density properties. It states that a measurable set "fills up" almost all of its points in a density sense, while its complement is sparse near almost all points outside the set.

ExampleDensity of Intervals

For an interval $I = (a, b)$ :

For any $x \in (a, b)$ : When $h$ is small enough, $(x-h, x+h) \subseteq (a, b)$ , so $d(I, x) = 1$
For $x \notin [a, b]$ : The intersection $I \cap (x-h, x+h)$ has measure $o(h)$ , giving $d(I, x) = 0$
At endpoints $x = a$ or $x = b$ : The density is $1/2$

This illustrates why "almost every" is necessary: the density can differ from $0$ or $1$ at boundary points, but these form a set of measure zero.

Remark

The Lebesgue Density Theorem has important consequences:

Lebesgue points: A point $x$ where $d(E, x) = 1$ is called a point of density or Lebesgue point of $E$ .
Connection to differentiation: The theorem is closely related to the Lebesgue Differentiation Theorem, which generalizes this to functions rather than just characteristic functions of sets.
Geometric measure theory: The density theorem extends to $\mathbb{R}^n$ and forms a cornerstone of geometric measure theory.

The proof relies on the Hardy-Littlewood maximal function and covering lemmas, particularly the Vitali Covering Lemma. The theorem demonstrates a deep connection between measure theory and real analysis: measurable sets have a regularity property that makes them behave like "nice" sets at most points, even though they can be quite complicated globally.