  We study the space BMO in the general setting of a measure space $\mathbb{X}$
with a fixed collection $\mathscr{G}$ of measurable sets of positive and finite
measure, consisting of functions of bounded mean oscillation on sets in
$\mathscr{G}$. The aim is to see how much of the familiar BMO machinery holds
when metric notions have been replaced by measure-theoretic ones. In
particular, three aspects of BMO are considered: its properties as a Banach
space, its relation with Muckenhoupt weights, and the John-Nirenberg
inequality. We give necessary and sufficient conditions on a decomposable
measure space $\mathbb{X}$ for BMO to be a Banach space modulo constants. We
also develop the notion of a Denjoy family $\mathscr{G}$, which guarantees that
functions in BMO satisfy the John-Nirenberg inequality on the elements of
$\mathscr{G}$.
