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Let $A$ be a compact metric space, and $E\subset A$ be a Borel measurable set. Does there exist a finite positive measure $\mu$ on the Borel sets of $A$ such that $\mu(E)>0$ and $\mu(A\setminus E)=0$?

The question is related to this question, except that we don't ask $E$ to be the support of $\mu$.

John
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    Certainly not if $E=\emptyset$, but given any $y\in E$ there's $\mu:\mathcal P(A)\to {0,1}$, $\mu(X)=\begin{cases} 1&\text{if }y\in X\ 0&\text{if }y\notin X\end{cases}$. – Sassatelli Giulio Mar 31 '24 at 01:36
  • Thank you for such a simple and nice observation. – John Mar 31 '24 at 01:38

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