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Most of what I've read defines the support of a distribution topologically if it defines support precisely; e.g the smallest closed subset that has full measure (i.e. the intersection of all closed subsets of full measure). I've seen allusions to purely measure-theoretic definitions, and I think I even came across one once, but I can't find it. Is there a measure-theoretic definition of support that everyone agrees on?

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Here is a comment made by Dmitri Pavlov for this MathOverFlow question. Maybe it will be helpful to you.

The support of a measure $m$ on a measurable space is defined as follows. Consider the Boolean algebra of all measurable sets modulo null sets (i.e., two measurable sets are equivalent if their symmetric difference is a measure $0$ set). This Boolean algebra is complete, as described in the link. Now take the supremum of all elements $p$ of this algebra such that $m$ vanishes on $p$. The complement of this supremum is the support of $m$. It is extremely important to factor out the null sets, otherwise the above procedure doesn't make sense. – Dmitri Pavlov Oct 16 at 15:09

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I'm not sure the support is interesting in a measure algebra. If you work with a normed measure algebra (a probability space modulo null sets), you get a unique element with measure $1$. In that context, you don't need a support. –  Michael Greinecker Dec 31 '11 at 13:57

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