Abstract We study the space BMO𝒢 (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 The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones.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 BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.

中文翻译：

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BMO 和测度空间上的 John-Nirenberg 不等式

摘要 我们在测量空间 𝕏 的一般设置中研究空间 BMO𝒢 (𝕏) 具有固定集合 𝒢 的可测量的正和有限测度集，由 𝒢 中集合上的有界平均振荡函数组成。目的是看看当度量概念被测度理论取代时，熟悉的 BMO 机制有多少存在。特别地，考虑了 BMO 的三个方面：它作为 Banach 空间的特性、它与 Muckenhoupt 权重的关系以及 John-Nirenberg 不等式。我们给出了 BMO𝒢 (𝕏) 为 Banach 空间模常数的可分解测度空间 𝕏 的充要条件。我们还开发了 Denjoy 族 𝒢 的概念，它保证 BMO𝒢 (𝕏) 中的函数满足关于 𝒢 元素的 John-Nirenberg 不等式。