Exercise 4, § 3

Notations and hypotheses are those of the preceding exercise. One calls abstract measure on $(E, \mathscr{T})$ every mapping m of $\mathscr{T}$ into $\overline{\mathbf{R}}_+$ having the following property: for every sequence $(A_n)_{n \in \mathbf{N}}$ of sets belonging to $\mathscr{T}$, pairwise disjoint, one has $m(\bigcup_{n \in \mathbf{N}} A_n) = \sum_{n \in \mathbf{N}} m(A_n)$. One calls integral on $(E, \mathscr{T})$ every mapping J of $\mathscr{M}$ into $\overline{\mathbf{R}}_+$ satisfying the following conditions: $\alpha)$ $J(\lambda \cdot f) = \lambda \cdot J(f)$ for $\lambda \in \overline{\mathbf{R}}_+$ and $f \in \mathscr{M}$ (with the usual convention $0 \cdot (+\infty) = 0$); $\beta)$ $J\left( \sum_{n \in \mathbf{N}} f_n \right) = \sum_{n \in \mathbf{N}} J(f_n)$ for every sequence $(f_n)_{n \in \mathbf{N}}$ of elements of $\mathscr{M}$.

a) Let J be an integral on $(E, \mathscr{T})$; for every subset $A \in \mathscr{T}$, set $m_J(A) = J(\varphi_A)$; show that $m_J$ is an abstract measure and that the mapping $J \mapsto m_J$ is a bijection of the set of integrals on $(E, \mathscr{T})$ onto the set of abstract measures on $(E, \mathscr{T})$ (if m is an abstract measure, first define $J(f)$ by linearity for the finite positive $\mathscr{T}$-step functions and use Exer. 3 e)). If m is an abstract measure, we will denote by $m^*$ the corresponding integral.

b) Let m be an abstract measure on $(E, \mathscr{T})$. For every positive function f on E, finite or not, set $m^*(f) = \inf_{g \in \mathscr{M}, g \geq f} m^*(g)$. Show that $m^*$ is an encumbrance on E (imitate the proof of Th. 3 of Ch. IV, §1, No. 3). One sets $m^*(A) = m^*(\varphi_A)$ for every subset A of E.