Exercise 9, § 3

Let $X$ be a set; one calls gauge on $X$ every countably additive mapping of $\mathscr{P}(X)$ into $\mathbf{R}_+$; a gauge $m$ on $X$ is said to be diffuse if $m(\{x\}) = 0$ for all $x \in X$, and to be atomic if there exists a positive function $f$ on $X$ such that $m(A) = \sum_{x \in A} f(x)$ for every subset $A$ of $X$.

a) Show that every gauge on $X$ may be decomposed in a unique way as the sum of a diffuse gauge and an atomic gauge.

b) Let $X'$ be a subset of $X$, and $m'$ a gauge on $X'$; for every subset $A$ of $X$, one sets $m(A) = m'(A \cap X')$. Show that $m$ is a gauge on $X$ and that it is diffuse (resp. atomic) if and only if $m'$ has the same property.

c) Let $m$ be a gauge on $X$ and let $(X_i)_{i \in I}$ be a family of pairwise disjoint subsets of $X$. If every gauge on $I$ is atomic, then $m(\bigcup_{i \in I} X_i) = \sum_{i \in I} m(X_i)$.