# Vitali theorem

Vitali's covering theorem. If a system of closed sets $\mathcal F$ is a Vitali covering (see below) of a set $A\subset\mathbb R^n$, it is possible to extract from $\mathcal F$ an at most countable sequence of pairwise disjoint sets $\{F_i\}$, $i=1,2,\dots$, such that \begin{equation} m_e\left[A\setminus\bigcup_{i=1}^{\infty}F_i\right]=0, \end{equation} where $m_e$ is the outer Lebesgue measure in $\mathbb R^n$.
A Vitali covering of a set $A\subset\mathbb R^n$ is a system $\mathcal E$ of subsets of $\mathbb R^n$ such that for any $x\in A$ there exists a sequence $\{E_n\}$ from $\mathcal E$ satisfying the following conditions: \begin{equation} x\in\bigcap_{n=1}^{\infty}E_n; \end{equation} \begin{equation} \delta_n = \delta(E_n) \to 0\quad \text{ if } n\to\infty, \end{equation} where $\delta(E_n)$ is the diameter of $E_n$; and \begin{equation} \inf_n\left[\sup\frac{m_e(E_n)}{m(I)}\right]=\alpha>0, \end{equation} where the supremum is taken over all $I$ (cubes with faces parallel to the coordinate planes and containing $E_n$), this supremum is said to be the regularity parameter of $E_n$.
The theorem was demonstrated by G. Vitali  for the case when $\mathcal F$ consists of cubes with faces parallel to the coordinate planes. Vitali's theorem is valid as stated if $\mathcal F$ is a Vitali covering of the set $A$ and not for a covering in the ordinary sense. This condition must always be satisfied, even if $\mathcal F$ is a system of segments and if to each $x\in A$ there corresponds a sequence $\{F_n\}$ from $\mathcal F$ with centres at $x$ and with diameters tending to zero.