Remark 2

Let us show that the measures introduced in Example 2 of No. 3 are measures with compact support in $T$. Let $\mu$ be a positive measure on $T$ whose support is a compact set $K$, and let $\nu$ be the measure defined by $\mu_K$ (in the sense of No. 3). Let $f \in \mathscr{F}_+(T)$; then

The encumbrance $\mu^\bullet$ being concentrated on $K$, we also have

whence $\mu^\bullet = \nu^\bullet$, and finally $\mu = \nu$. Conversely, if $K$ is a compact set in $T$ and $\lambda$ a measure on $K$, and if $\mu$ is the measure on $T$ defined by $\lambda$, then $\mu^\bullet(\mathbf{C}K) = 0$ (No. 3, formula (3)); consequently, the support of $\mu$ is contained in $K$, hence is compact.