# 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.

Mathematics Asked on November 18, 2021

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