Let $(\mathbb{R},M_\mu,\mu)$ where $\mu$ is Lebesgue-Stieltjes.
Let $E\subset \mathbb{R}$.
I want to show that the following are equivalent.
a. $E \in M_\mu$
b. $E=V \setminus N_1$ where $V$ is a $G_\delta$ set and $\mu (N_1) =0$.
c. $E=H\bigcup N_2$ where $H$ is an $F_\sigma$ set and $\mu (N_2)=0$.
I Show that this proper when $\mu(E) < \infty$ through Folland's book.
How Can do that when $\mu(E)$ has infinity.
First I have tried..
$E_j = E \cap (j,j+1]$ and find $G_\delta$ set $V_j$ and $N_j$ such that $E_j = V_j \setminus N_j$
and compute $\bigcup E_j$.
What shall I do?
Let $E \in M_{\mu}$ be given and put $F = \Bbb R \setminus E$. Write $F = \bigcup F_j$ where $F_j = [-j,j] \cap F$. We have $(F_j) \subset M_{\mu}$ and $\mu F_j < \infty$ for all $j$, so by the previous case, for each $j$ there is an $F_{\sigma}$ set $H_j$ and a negligible set $N_j$ such that $F_j = H_j \cup N_j$. Then, $F = \bigcup F_j = (\bigcup H_j) \cup (\bigcup N_j) = H \cup N$ where $H = \bigcup H_j$ is $F_{\sigma}$ as the countable union of $F_{\sigma}$'s and $N = \bigcup N_j$ is clearly negligible.
Therefore, $E = \Bbb R \setminus F = (\Bbb R \setminus H) \setminus N = V \setminus N $, where $V = \Bbb R \setminus H$ is $G_{\delta}$ as the complement of an $F_{\sigma}$.
This proves $b.$ but implicitly proves $c.$ as well.