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I'm studying Measure Theory and have recently come across Lusin's Theorem which states:

Let $f$ be (Lebesgue) measurable on $[a,b]$. Given $\epsilon>0$, there exists a continuous function $g$ s.t. $m( \{ x : f(x) \neq g(x) \} ) < \epsilon $ and $\sup |g|< \sup |f| $

However, I haven't come across any applications or consequences of this. What are some important applications of Lusin's Theorem?

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    $\begingroup$ With this theorm you prove that every measurble function is a limit of continuous functions such that they agree on more and more points. And there are many more applications. $\endgroup$
    – Yonatan
    Commented Mar 19, 2016 at 14:38
  • $\begingroup$ But can you tell me a question or a theorem where we can apply it? $\endgroup$ Commented Mar 20, 2016 at 13:50
  • $\begingroup$ You can prove that $$\lim_{\alpha \ to 0} { \int_{\Omega}{(f(x+\alpha)-f(x))dx}} =0$$ $\endgroup$
    – Yonatan
    Commented Mar 20, 2016 at 19:05

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As others have pointed out, you can use Lusin's theorem to prove that the space $C_c([a,b])$ of complex continuous functions with compact support is dense in $L^p([a,b])$ for all $p\in [1,\infty)$. (For a more general statement, replace $[a,b]$ by a locally compact Hausdorff space $X$, and the Lebesgue measure by some Borel measure induced by the Riesz Representation theorem).

In practice, we want to work with continuous functions, which behave nicely and allow many operations. But the space of continuous functions is not complete, which means we are not guaranteed to stay in it. Hence we start with operations on continuous functions (or maybe even smooth ones), and extend them to $L^p$ functions. For many an operation (a bounded one), $C_c$ being dense in $L^p$ ensures the uniqueness of its extension. I cannot address how important Lusin's theorem is in this regard.

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  • $\begingroup$ Would you please direct me to an online source that uses Lusin's theorem to prove that $C_c(E)$ is dense in $L^p(E)$ where $E$ is a measurable subset of the real numbers. $\endgroup$
    – user0
    Commented Jan 15, 2022 at 18:54

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