Proposition 2

a) Let w be a positive premeasure. For every compact subset K of T, the encumbrance $(w^\bullet)_K$ induced by $w^\bullet$ on K is equal to $(w_K)^\bullet$. For every function $f \in \mathscr{F}_+(T)$, one has the relations $(w_K)^\bullet(f_K) = w^\bullet(f_{\varphi_K})$ and

*b) Conversely, let p be an encumbrance on T satisfying the following conditions: 1) For every compact subset K of T, there exists a positive measure $w_K$ on K such that $p_K = (w_K)^\bullet$. 2) For every function $f \in \mathscr{F}_+(T)$, $p(f) = \sup_K p(f_{\varphi_K})$. The mapping $w : K \mapsto w_K$ is then a positive premeasure on T, and $p = w^\bullet$.

Let us prove $a$: let $g \in \mathscr{F}_+(K)$ and let $g^0$ be the extension by zero of $g$ to $T$; then, by the definition of induced encumbrances,

where $L$ runs over the set of compact subsets of $T$, or merely the set of those that contain $K$. But if $L$ contains $K$, then $(w_L)^\bullet(g^0|L) = (w_K)^\bullet(g)$ from the fact that $g^0|L$ is zero outside $K$ (Ch. V, §7, No. 1, Prop. 1), which proves the first assertion. Therefore

for all $f \in \mathscr{F}_+(T)$, and (2) merely translates the formula (1).

Let us pass to $b$: the measure $w_K$ considered in 1) is unique (Ch. V, §1, No. 1). Let us show that the mapping $K \mapsto w_K$ is a premeasure: let $K$ and $L$ be two compact subsets such that $K \subset L$, and let $\lambda$ be the measure induced by $w_L$ on $K$; everything comes down to showing that $\lambda^\bullet = (w_K)^\bullet$. Now, $\lambda^\bullet = ((w_L)^\bullet)_K$ (Ch. V, §7, No. 1, Prop. 1); since $(w_L)^\bullet = p_L$, we have $\lambda^\bullet = (p_L)_K = p_K = (w_K)^\bullet$.

Let us denote by $w$ the premeasure $K \mapsto w_K$; since $p_K = (w_K)^\bullet = (w^\bullet)_K$, we have $p(f_{\varphi_K}) = p_K(f_K) = (w^\bullet)_K(f_K) = w^\bullet(f_{\varphi_K})$. The two encumbrances $p$ and $w^\bullet$ are therefore equal by virtue of the formula (2) and the hypothesis 2) on $p$.

Q.E.D.

Since the induced encumbrance $(w^\bullet)_K$ is equal to $(w_K)^\bullet$, there is no ambiguity in simply writing $w_K^\bullet$. We shall employ this notation from now on.