Strong and Weak Vitali Properties

Abstract

Let $X$ be a metric space and let $\mu$ be a Borel measure on $X$. We say that $\mu$ satisfies the strong Vitali property if for any Borel subset $E$ of $X$ with $\mu (E) <\infty$ and for any fine cover $\mathcal{V}$ of $E$, we may extract a countable disjoint subcollection $\pi = \{B_{i}\}$ from $\mathcal{V}$ such that $\displaystyle \mu(E \setminus \cup B_{i}) = 0$. If we require $\pi$ to satisfy the condition that if $B(x_{i},r_{i}), ~B(x_{j},r_{j}) \in \pi$ with $i \neq j$, then $x_{i} \notin B(x_{j},r_{j})$, then $\mu$ is said to satisfy the weak Vitali property. Besicovitch showed that every finite Borel measure in $\mathbb{R}^{n}$ must satisfy the strong Vitali property. It is also true for certain other metric spaces. In a general metric space it is no longer pertinent to ask if all Borel measures possess a certain property. To this end we construct a metric space $\Omega$ and identify two subsets $A, B \subseteq \Omega$ such that for any Borel probability measure $\mu$ on $\Omega$,

1.$\mu (A) = 1$, then $\mu$ must satisfy the strong Vitali property;

2. $\mu(B) = 1$, then $\mu$ must satisfy the weak Vitali property but not necessarily the strong Vitali property;

We introduce the notion of a centralized Vitali property and give an example of a measure for which this property fails.