BMO and the John-Nirenberg Inequality on Measure Spaces

We study the space BMO () in the general setting of a measure space with a fixed collection of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in . The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones.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 for BMO () to be a Banach space modulo constants. We also develop the notion of a Denjoy family , which guarantees that functions in BMO () satisfy the John-Nirenberg inequality on the elements of