Exercise 5, § 1

Let T be a set. Denote by $\mathscr{B}_+(T)$ the set of bounded positive real functions on T, and by $p_0$ a mapping of $\mathscr{B}_+(T)$ into the interval $[0, +\infty]$ of $\overline{\mathbf{R}}$ satisfying the conditions a) to d) of Def. 1 of No. 1. For every function $f \in \mathscr{F}_+(T)$, denote by $p(f)$ the supremum of the set of numbers $p_0(f_0)$, where $f_0$ runs over the set of bounded functions $\leq f$. Show that $p$ is an encumbrance on T. Analogous question, with $\mathscr{B}_+(T)$ replaced by the set of (finite) positive real functions on T.