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?