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?

  • $\begingroup$ This is a basic theorem in the theory of measure and integration. Try to read the proof from your favorite book and if you get stuck we would be glad to help you. $\endgroup$ May 21 '19 at 6:20
  • $\begingroup$ Perhaps a relevant post here $\endgroup$
    – Wei Zhong
    May 21 '19 at 6:45
  • $\begingroup$ @KaviRamaMurthy Could you please recommend a text? I have read Rudin and Royden but they don't have this theorem $\endgroup$
    – Bunbury
    May 21 '19 at 8:13
  • $\begingroup$ They both have this result. In Rudin See the section "Approximation by continuous functions" in the chapter on $L^{p}$ spaces. $\endgroup$ May 21 '19 at 8:16

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