Let $K=[0,1]$, $\mathfrak{R}$ is its usual Borel sigma field. $\mu$ is Lesbeque measure, $S= K\setminus \mathbb{Q}$. Then, for all Baire sets $E \subset \mathbb{Q}, \mu(E)=0$, while $$\text{supp}(\mu)= K \ne S$$

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 $$\text{supp}(\mu)= K \ne S$$