Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.
Since the question is asking for a sequence of continuous functions that approximate $f$ in the integral, I'm thinking about applying Lusin's theorem and the monotone convergence theorem. Construct a sequence of increasing continuous functions $\{f_n\}$ such that $f_n \to f$. Then $\int_X f_n d\mu \to \int_X f d\mu$. Am I on the right track?
For example, let $f_n=f-\frac{1}{n}$. We have $\int_X f_n d\mu \to \int_X f d\mu$. By Lusin's theorem, there exists a continuous function $g_n$ such that $\mu(\{x|f_n(x)\neq g_n(x)\})<\epsilon$. But how can I relate $g_n$ to $f$ in the last step?
