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Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure described "bounded" sets instead of "open" sets. Similarly, the morphisms are "bounded" instead of "continuous".

Formally, that a bornological space is a pair $(X, \mathbf B)$, where $\mathbf B$ is a set of subsets of $X$ which covers $X$, is downward-closed, and is closed under finite unions [wiki | nLab]. We may trivially assign a measurable structure to every bornological space by defining simply $\mathcal B = \sigma(\mathbf B)$, the minimal $\sigma$-algebra which contains the bornology $\mathbf B$.

What kind of measure theory can one do in this setting? For example, are there necessary and sufficient conditions for a bornological space to admit a non-trivial measure? Is there a classification of $\sigma$-ideals for a bornological space?

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Since it is downward-closed, you get lots of power-sets contained in your sigma-algebra. So is this a question on measurable cardinals? –  Gerald Edgar Feb 27 '13 at 14:32

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Usually we find a measure space and look into its bornological stuff, not conversely, a helpful reference may be founded in this book: U. H¨ohle and S. E. Rodabaugh, eds., Mathematics of fuzzy sets: logic, topology and measure theory, Handbook Series Another reference may be S. T. Hu's research papers.

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