Exercise 2, § 3

Let E be a set, $\Phi$ a clan of subsets of E, and V a fully lattice-ordered Riesz space. A mapping m of $\Phi$ into V is said to be positive if $m(A) \geq 0$ for all $A \in \Phi$, that it is additive if $m(A \cup B) = m(A) + m(B)$ when the sets $A \in \Phi$ and $B \in \Phi$ are disjoint, and that it is relatively bounded if the set of elements $m(B)$ for $B \in \Phi, B \subset A$ is bounded above in V for any $A \in \Phi$.

a) Let m be a relatively bounded mapping of $\Phi$ into V. One defines for every A the elements $m_+(A), m_-(A)$ and $|m|(A)$ of V as in Exer. 1 a). Show that $|m|(A)$ is the supremum of the set of elements of V of the form $\sum_{i=1}^{p} |m(A_i)|$ for all the finite partitions $(A_i)_{1 \leq i \leq p}$ of A into sets belonging to $\Phi$. From this, deduce that the mapping $|m|$ of $\Phi$ into V is additive, then that m is the difference of the additive positive mappings $m_+$ and $m_-$ of $\Phi$ into V.

b) Let $m'$ and $m''$ be two additive positive mappings of $\Phi$ into V and let $m = m' - m''$. Show that m is relatively bounded, and that $m' \geq m_+$ and $m'' \geq m_-$.

c) Generalize Exer. 1 d).