We study the space BMO G (X) in the general setting of a measure space X with a fixed collection G of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 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 The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones.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 X for BMO G (X) to be a Banach space modulo constants. We also develop the notion of a Denjoy family G , which guarantees that functions in BMO G (X) satisfy the John-Nirenberg inequality on the elements of G .2010 Mathematics Subject Classification. Primary 30L15 42B35 46E30. G.D. was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Centre de recherches mathématiques (CRM). R.G. was partially supported by the Centre de recherches mathématiques (CRM), the Institut des sciences mathématiques (ISM), and the Fonds de recherche du Québec -Nature et technologies (FRQNT). A.L.
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