Exercise 2, § 1

Let T be a set and p an encumbrance on T. A function $f \in \mathscr{F}_+(T)$ (resp. a subset A of T) is said to be p-negligible if $p(f) = 0$ (resp. $p(A) = 0$). a) Let f and g be in $\mathscr{F}_+(T)$ and let t be a positive real number such that $f \leq t \cdot g$. If g is p-negligible, then so is f. The sum and upper envelope of a sequence of p-negligible functions are p-negligible. b) Every subset of a p-negligible set is p-negligible, as is the union of a sequence (finite or infinite) of p-negligible sets. c) Let $f \in \mathscr{F}_+(T)$. For every finite real number $a > 0$, denote by $T_a$ the set of $t \in T$ such that $f(t) \geq a$. Show that $p(T_a) \leq a^{-1} \cdot p(f)$. From this, deduce that f is p-negligible if and only if the set of $t \in T$ such that $f(t) \neq 0$ is p-negligible. d) Let $f \in \mathscr{F}_+(T)$. If $p(f)$ is finite, the set of $t \in T$ such that $f(t)$ is infinite is p-negligible. e) Let f and g be in $\mathscr{F}_+(T)$. If the set of $t \in T$ such that $f(t) \neq g(t)$ is p-negligible, then $p(f) = p(g)$.