Let $(\Bbb R,\mathcal A_{\Bbb R}^*,\bar\lambda)$ be the complete Lebesgue measure space. Let $f:\Bbb R\to\Bbb R$ be a function. Fréchet's theorem states that:
$$f\in M(\Bbb R,\mathcal A_{\Bbb R}^*)\Leftrightarrow\exists\;\text{a sequence of continuous functions}\;\;(f_n)_{n\in\Bbb N}\;\text{s.t.}\;\;f_n\to f\;a.e.-\bar\lambda$$
Proof: $\mid\Rightarrow)$ part
Let's assume $f\in M(\Bbb R,\mathcal A_{\Bbb R}^*)$, by Lusin's theorem,we get that:
$$\forall n\in\Bbb N\;\exists\;E_n\;\text{open, with}\;\bar\lambda(E_n)<\frac{1}{n},\;\ \text{s.t.}\;\;f_n=f\mid_{E_n^c}\;\;\text{is continuous}$$
so letting $N=\bigcap_{n\in\Bbb N}E_n$ we get that $\bar\lambda(N)=0$ but know got stuck trying to prove that $\lim_{n\to\infty}f_n(x)=f(x)\;\forall x\in\Bbb R\setminus N$.
Any ideas would be appreciated.