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What is the difference between a measure being "supported on [set] $A$" vs "supported in $A$" vs "supported at $A$"?

What is the difference between saying three things (here $A$ is a set)

(1) Let $mu$ be a measure supported on $A$

and

(2) Let $mu$ be a measure supported in $A$

and

(3) Let $mu$ be a measure supported at $A$

I believe that (1) and (3) mean the same thing, which I think could be written for short as $operatorname{supp} mu= A$. If this is the case, then I think (2) would mean that $operatorname{supp} musubseteq A$.

But still, I feel like my understanding is not completely correct. I could not find any definition in books/websites of what saying "supported $dots$" means.

Please enlighten me. Thank you

Mathematics Asked on November 18, 2021

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