Exercise 3, § 3

Let E be a set and $\mathscr{T}$ a tribe of subsets of E. Denote by $\mathscr{M}$ the set of mappings f of E into $\overline{\mathbf{R}}_+$ such that the set of $x \in E$ for which $f(x) \geq c$ belongs to $\mathscr{T}$ for every $c \in \overline{\mathbf{R}}_+$.

a) Show that the limit superior and limit inferior of every sequence of elements of $\mathscr{M}$ belong to $\mathscr{M}$ (first study sequences that are increasing or decreasing).

b) Show that $\mathscr{T}$ is the set of subsets of E whose characteristic function belongs to $\mathscr{M}$.

c) Let $f \in \mathscr{M}$. For every Borel subset B of $\overline{\mathbf{R}}_+$, the set $f^{-1}(B)$ belongs to $\mathscr{T}$ (the set of subsets B of $\overline{\mathbf{R}}_+$ such that $f^{-1}(B) \in \mathscr{T}$ is a tribe, containing the intervals $[c, +\infty]$).

d) Let $f_1, \ldots, f_n$ be in $\mathscr{M}$ and let $\varphi$ be a Borel mapping of $(\overline{\mathbf{R}}_+)^n$ into $\overline{\mathbf{R}}_+$; show that the mapping $x \mapsto \varphi(f_1(x), \ldots, f_n(x))$ of E into $\overline{\mathbf{R}}_+$ belongs to $\mathscr{M}$. Deduce from this that $\mathscr{M}$ contains the sum of every series of elements of $\mathscr{M}$.

e) Show that $\mathscr{M}$ is the set of limits of increasing sequences of finite positive $\mathscr{T}$-step functions.