Let $(X, \mathcal{U})$ be a compact metric space and $\mu$ be a Borel probability measure such that $supp(\mu)=X$, $supp(\mu)$ is the support of $\mu$. Also take $\mathcal{A}^+_X= \{A\in B_X: \mu(A)>0\}$ where $B_X$ is the set of Borel sets of $X$.
What can say about relation $\mathcal{A}^+_X$ and $\mathcal{U}$? Is it true that $\mathcal{U}\subseteq \mathcal{A}^+_X$?
If $\mathcal U$ is supposed to be Borel sigma algebra the answer is NO. Consider Lebesgue measure on $[0,1]$ and the set $\{0\}$. This has measure $0$ even though the support of $\mu$ is $[0,1]$.
The support is $[0,1]$ because every non-empty set has positive measure.
If $\mathcal U$ is supposed to be the class of all open sets then an element $U$ of $\mathcal U$ belongs to $\mathcal A_X^{+}$ iff it is non-empty.