Definition 1

— Let T be a set. One calls encumbrance on T any mapping p of $\mathscr{F}_+(T)$ into $\overline{\mathbf{R}}_+$ that has the following properties: a) If f and g are two elements of $\mathscr{F}_+$ such that $f \leq g$, then $p(f) \leq p(g)$. b) If f is an element of $\mathscr{F}_+$, and t is a number $\geq 0$, then $p(tf) = tp(f)$.

c) If f and g are two elements of $\mathscr{F}_+$, then $p(f + g) \leq p(f) + p(g)$. d) If $(f_n)$ is an increasing sequence of elements of $\mathscr{F}_+$, and if $f = \lim_{n \to \infty} f_n$, then $p(f) = \lim_{n \to \infty} p(f_n)$. If A is a subset of T, we write $p(A)$ instead of $p(\varphi_A)$.

The condition b) implies that $p(0) = 0$. On the other hand, let $(f_n)$ be a sequence of elements of $\mathscr{F}_+$; the conditions c) and d) imply the inequality

(the inequality of countable convexity).

For example, let T be a locally compact space, $\mu$ a positive measure on T; then $\mu^*$ and $\mu^\bullet$ are encumbrances on T. This follows from Props. 10, 11, 12 and Th. 3 of Ch. IV, §1, No. 3 for $\mu^*$, and from Prop. 1 of Ch. V, §1, No. 1 for $\mu^\bullet$.