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Vitali Covering Theorem plays a key role in the differentiation theory of measures, particularly in proving Lebesgue's Differentiation Theorem, which states that the derivative of the integral exists almost everywhere.

Given a bounded set $E$ in $extbf{R}^n$ and a Vitali covering of $E$ by closed balls, there exists a disjoint subcollection of balls whose total measure differs from the measure of $E$ by at most $au$ for any given $au > 0$.

Vitali's Covering Theorem is used in the proof of Carathéodory's criterion for measurability, which characterizes the measurability of a set in terms of the outer measures of its coverings.

A collection of sets is called a Vitali covering of a set $E$ if, for every point $x$ in $E$ and every $au > 0$, there exists a set in the collection that contains $x$ and has diameter less than $au$.