Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of $K$ not necessarily in $\mathfrak{M}$. Suppose for all Baire sets $E\subseteq K\setminus S$, $\mu(E)=0$. Can I conclude that $Supp(\mu)\subset S$?



Support of a regular measure Reg
Let $K=[0,1]$, $\mathfrak{R}$ its usual Borel $\sigma$-field. and $\mu$ Lebesgue measure, $S= K\setminus \mathbb{Q}$.
Then, for all Baire sets $E \subset \mathbb{Q}, \mu(E)=0$, while $$\text{supp}(\mu)= K \ne S$$


