Example

— 1) For a measure on $X$ to be zero, it is necessary and sufficient that its support be empty. 2) The support of Lebesgue measure on $\mathbf{R}$ is the entire line $\mathbf{R}$; for, it is nonempty and is invariant under every translation.

3) On the interval $X = [0,1]$ of $\mathbf{R}$ consider a countable dense subset, arranged into a sequence $(a_n)$, and let $\mu$ be the measure defined by placing the mass $2^{-n}$ at the point $a_n$ for every $n \geq 0$ (§ 1, No. 3, Example I). The support of $\mu$ is all of $X$; for, let $x$ be any point of $X$, $V$ a neighborhood of $x$, and $f$ a continuous real-valued function $\geq 0$ on $X$, equal to 1 at the point $x$, whose support is contained in $V$ (§ 1, No. 2, Lemma 1); the set of $y \in V$ such that $f(y) > 0$ is open in $X$, therefore contains a point $a_n$, consequently $\mu(f) \geq f(a_n)2^{-n} > 0$.