Exercise 6, § 3

Let $E$ be a set, $\mathscr{T}$ a tribe of subsets of $E$, and $m$ an abstract measure on $(E, \mathscr{T})$. Suppose that $m(E)$ is finite and that, for every $B \in \mathscr{T}$ such that $m(B) > 0$, there exists a set $A \in \mathscr{T}$ such that $A \subset B$ and $0 < m(A) < m(B)$. For every $B \in \mathscr{T}$ and every number $t$ such that $0 \leq t \leq m(B)$, there exists a set $A \in \mathscr{T}$ such that $m(A) = t$ and $A \subset B$.