Differentiation of Measures - Applications

TheoremVitali Covering Theorem

Let $E \subseteq \mathbb{R}$ be a set and let $\mathcal{F}$ be a collection of closed intervals. We say $\mathcal{F}$ covers $E$ in the sense of Vitali if for every $x \in E$ and every $\epsilon > 0$ , there exists $I \in \mathcal{F}$ with $x \in I$ and $|I| < \epsilon$ .

If $E$ has finite outer measure and $\mathcal{F}$ covers $E$ in the sense of Vitali, then there exists a countable disjoint subcollection $\{I_n\} \subseteq \mathcal{F}$ such that: $\lambda^*\left(E \setminus \bigcup_{n=1}^{\infty} I_n\right) = 0$

In other words, a countable disjoint subcollection covers $E$ up to a set of measure zero.

The Vitali Covering Theorem is a fundamental tool for proving differentiation results. It ensures that "fine" coverings can be refined to countable disjoint coverings without losing essential coverage.

ExampleApplication to Density Points

Using Vitali's theorem, one can prove that for a measurable set $E \subseteq \mathbb{R}$ , almost every $x \in E$ is a density point: $\lim_{h \to 0} \frac{\lambda(E \cap (x - h, x + h))}{2h} = 1$

The proof uses Vitali to select intervals around points where density is not $1$ , showing such points form a negligible set.

Remark

Applications of the Vitali Covering Theorem:

Hardy-Littlewood maximal theorem: The Vitali lemma (a variant) is key to proving: $\lambda(\{x : Mf(x) > \alpha\}) \leq \frac{C}{\alpha} \int |f| \, d\lambda$

where $Mf$ is the maximal function.

Lebesgue differentiation: For $f \in L^1_{\text{loc}}(\mathbb{R}^n)$ : $\lim_{r \to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y) \, dy = f(x) \text{ a.e.}$

Vitali's theorem helps show that exceptional sets have measure zero.

Differentiation of signed measures: If $\mu$ is a finite signed measure on $\mathbb{R}$ , then: $\lim_{h \to 0} \frac{\mu([x - h, x + h])}{2h}$

exists for almost every $x$ and equals the Radon-Nikodym derivative $\frac{d\mu}{d\lambda}(x)$ where $\mu \ll \lambda$ .

Besicovitch covering theorem: A more refined version for $\mathbb{R}^n$ that allows overlaps but controls the multiplicity. This is essential in geometric measure theory.
Differentiation of BV functions: The Vitali theorem is used to prove that functions of bounded variation are differentiable almost everywhere.

The Vitali Covering Theorem exemplifies how geometric arguments (covering sets with intervals) lead to measure-theoretic conclusions, bridging geometry and analysis in profound ways.