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If $X$ is a metric space and $B$ is the Borel $\sigma$-algebra and $\mu$ is a measure on $(X,B)$.Define support of $\mu$ as the smallest closed set $F$ such that $\mu(F^{c})=0$. Let $X=[0,1]$. Show that if $F$ is a closed subset of $X$, then there exists a finite measure on $X$ whose support is $F$.

I thought that the obviuos choice to construct such a measure is as follows : $$\mu(A)=m(A\cap F)$$.

Where m is the lebesgue measure on $[0,1]$. But this example doesn't seem to work.

Thanks in advance for any help.

Riju
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  • @DominiqueR.F. I have already seen that post, but I didn't find it helpful.. – Riju May 12 '17 at 14:52
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    Then try editing your question to make it more specific. For example, you could put a link to the older post and ask specific questions about the steps that you didn't understand in the answers given there. If you leave your question as it is now, it will probably be closed as a duplicate. – Derived Cats May 12 '17 at 15:01

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