I was wondering if there is some definition for "support" of a measure in the sense that one or both of the following can be true:
- one measure is absolutely continuous with respect to another measure, if and only if the support of the former is inside the support of the latter?
- two measures are mutually singular (as in Rudin's Real and Complex Analysis), if and only if the supports of the two measures are disjoint?
- The definition for support of a measure in Wikipedia relies on that the measurable space is also a topological space. I would like to know if it makes sense to define support of a measure solely on a measurable space?
Thanks and regards!