Definition 2

Let p be an encumbrance on a set T. One says that p is bounded if $p(T) < +\infty$. If T is a topological space, p is said to be locally bounded provided that every $x \in T$ admits a neighborhood V such that $p(V) < +\infty$.

It then follows from the properties a) and c) of Def. 1 that $p(K) < +\infty$ for every compact subset K of T. In particular, if T is compact, then every locally bounded encumbrance on T is bounded.

Let p be an encumbrance on a set T, and A a subset of T. For every function $f \in \mathscr{F}_+(A)$, let $f^0$ be the extension by 0 of f to T; the mapping $f \mapsto p(f^0)$ on $\mathscr{F}_+(A)$ is then an encumbrance, called the encumbrance induced by p on A, and is denoted $p|A$ or $p_A$.

Let T and U be two sets, $\pi$ a mapping of T into U, and p an encumbrance on T. The encumbrance $\pi(p)$ on U, whose value for $f \in \mathscr{F}_+(U)$ is given by

is called the image encumbrance of $p$ under $\pi$.

Let $p$ be an encumbrance on a set $T$; $p$ is said to be concentrated on a subset $A$ of $T$ if $p(T - A) = 0$.