If $A\in\mathfrak{M}$, then there exist Borel sets $F$ and $G$ such that $F\subset A\subset G$, and $\mu(G-A)=\mu(A-F)=0$.

Question

The problem arises from Walter Rudin's Principles of Mathematical Analysis:

Here is (b):

Now, the highlighted texts seem innocent enough; but I failed to prove it rigorously, i.e., to explicitly show the existence of $F$ and $G$. Any hint would be greatly appreciated.

Answer 1

If we use (b), we take a sequence of open sets $A \subset G_{k}$ and a sequence of closed sets $F_{k}\subset A$ such that $\mu(G_{k}-F_{k})<\frac{1}{k}$ and we define $G=\bigcap_{k}G_{k}$ and $F=\bigcup_{k}F_{k}$ and these are the desired sets.