Existence of a measurable set , $B\supseteq A$ with $\mu^\ast(A)=\mu^\ast(B)$

Question

Let $A\in \mathbf{R}$. How can I show that $\exists$ a measurable set $B\supseteq A$ such that $\mu^\ast(A)=\mu^\ast(B)$?

I know there is a $G_\delta$ set such that $\mu^\ast(A)=\mu^\ast(G)$. Can I take $B$ to the a $G_\delta$?