Corollary 2

— Let $\nu$ be a real measure on $T$. In order that $\nu$ belong to the band generated by $\mu$ in the fully lattice-ordered space $\mathscr{M}(T)$ (Ch. II, §1, No. 5), it is necessary and sufficient that $\nu$ be a measure with base $\mu$.

On considering $\nu^+$ and $\nu^-$, one is immediately reduced to the case of a positive measure $\nu$ (No. 2, Cor. of Prop. 2). Let us then set $\nu_n = \inf(n\mu, \nu)$; $\nu$ belongs to the band generated by $\mu$ if and only if $\nu = \sup_n \nu_n$ (Ch. II, §1, No. 5, Cor. of Prop. 6). Now $\nu_n$, being bounded above by $n\mu$, is a measure with base $\mu$ by Th. 2; the relation $\nu = \sup_n \nu_n$ therefore implies that $\nu$ is a measure with base $\mu$ (Cor. 1). Conversely, suppose that $\nu$ is a measure with base $\mu$: $\nu = g \cdot \mu$, where $g$ is locally $\mu$-integrable and positive. Then $\nu_n = \inf(g, n) \cdot \mu$ (Cor. of Prop. 2), and it follows at once from Lebesgue’s theorem (Ch. IV, §4, No. 3, Prop. 4) that $\nu = \sup_n \nu_n$.