Lebesgue's density theorem

Remarks

Apply the Lebesgue differentiation theorem to the indicator function $\mathbf{1}_{A \cap B(0, R)}$ for each integer $R$. Its ball averages are exactly the ratios in the statement, and its values are $1$ on $A \cap B(0,R)$ and $0$ off it.

Example: for $A = [0, \infty) \subseteq \R$, the density is $1$ at every $x > 0$, $0$ at every $x < 0$, and $1/2$ at $x = 0$. The exceptional set $\{0\}$ has measure zero, as the theorem allows.

The theorem says that a measurable set cannot be spread out evenly at small scales: near almost every one of its points it fills almost all of a small ball. For instance, no measurable $A \subseteq [0,1]$ can satisfy $\lambda(A \cap I) = \lambda(I)/2$ for every interval $I \subseteq [0,1]$.

Going deeper

For a student

The points where the density is $1$ are called density points. The density theorem is used to show that measurable sets are, up to null sets, sets of density points, which leads to the density topology on $\R$. It can also be used to prove the Steinhaus theorem that $A - A$ contains an interval around $0$ when $\lambda(A) > 0$.