Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ by
$ B(U;r) =$ { $ V\in\Sigma : \mu(U \cup V) < r $ }
So a $\mu$-ball is a set of sets. Then finding suitable properties of the collection of all $\mu$-balls on $(X, \Sigma, \mu)$ to use as axioms for a measure-space analogy of topological spaces, where such spaces that didn't arise from a measure-space would be called non-measurizable.
The above definition of a $\mu$-ball is just a simple analogy from an open ball in a metric space, however perhaps a totally different way of generalizing a measure space would more naturally relate to measure spaces as topological spaces relate to metric spaces.
If connectedness in topological spaces is similar to connectedness in graphs then perhaps connectedness in measure-space analogy of topological spaces is similar to connectedness in hypergraphs.
