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1. 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?

2. 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!

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1 Answer

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1. No way: Think of Lebesgue measure and a point measure.

2. You can often speak of the support up to a null-set, but in order to single out a specific support you need further structure on your measure space.

Added: In 2. I was deliberately a bit sloppy. A sufficient condition for the existence of a good notion of support is a class of null sets closed under arbitrary unions. The union of those null-sets is then the largest such $\mu$-null set and its complement deserves the name of support of $\mu$. For instance, if $\mu$ happens to be a Radon measure on a locally compact space, you can take the class of open $\mu$-null sets and the union of those is precisely the complement of the (closed) support of $\mu$.

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