Exercise 1, § 3

Let E be a set, $\Phi$ a clan of subsets of E, and m an additive set function defined on $\Phi$. Denote by $\mathscr{E}(\Phi)$ the vector space of real $\Phi$-step functions on E, and by J the linear form on $\mathscr{E}(\Phi)$ such that $J(\varphi_A) = m(A)$ for all $A \in \Phi$.

a) For every $A \in \Phi$, one sets

Show that $|m|(A) = m_+(A) + m_-(A)$ is the supremum of the set of numbers 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 $\Phi$.

b) One says that m is relatively bounded if the set of numbers of the form $m(B)$ with $B \in \Phi, B \subset A$ is bounded for any set $A \in \Phi$. Show that the following conditions are equivalent: $\alpha)$ m is relatively bounded; $\beta)$ the linear form J on the Riesz space $\mathscr{E}(\Phi)$ is relatively bounded; $\gamma)$ m is the difference of two positive additive set functions on $\Phi$; $\delta)$ $|m|(A)$ is finite for all $A \in \Phi$ (argue in a circle $\alpha$) $\Rightarrow \beta) \Rightarrow \gamma) \Rightarrow \delta) \Rightarrow \alpha)$.

c) Suppose that m is relatively bounded. Show that the set functions $|m|, m_+$ and $m_-$ are additive and that $m = m_+ - m_-$ (make use of Th. 1 of Ch. II, §2, No. 2). Moreover, if $m'$ and $m''$ are additive and positive set functions on $\Phi$ such that $m = m' - m''$, then $m' \geq m_+$ and $m'' \geq m_-$.

d) One says that m is countably additive if $m(A) = \sum_{n \geq 1} m(A_n)$ for every countable partition $(A_n)_{n \geq 1}$ of a set $A \in \Phi$ into sets belonging to $\Phi$. Show that this condition signifies that $\lim_{n \to \infty} m(A_n) = 0$ for every decreasing sequence $(A_n)_{n \geq 1}$ of elements of $\Phi$ such that $\bigcap_{n \geq 1} A_n = \emptyset$. If m is countably additive, show that $|m|$ is countably additive.