I have a question regarding Lebesgue-Integration. I am working with the book by Ziemer (Modern Real Analysis, https://www.math.purdue.edu/~torresm/pubs/Modern-real-analysis.pdf) and I am trying to solve exercise 2 on page 162.

Ziemer uses the the definition of the Lebesgue-integral which is described here or in the link above: Alternative integral definition (But with the correction that for the integral to be existent it is supposed that $f$ is measureable.)

Now I have to show that if $$\underline \int f = \overline \int f <\infty$$ then there exists an integrable fuction $g$ s.t. $f=g$ a.e. So I have to show, that there is a measurable function $g$ s.t $\underline \int g = \overline \int g <\infty$.

I tried to work with the definition of the integrals: I showed, that for every $n\in\mathbb{N}$ there are measurable functions $k_n\leq f\leq h_n$ with $$ \int h_n - \int k_n < 1/n. $$ Taking $k:=\sup_n k_n$ and $h:=\inf_n h_n$ I get measurable functions and $k\leq f\leq h$. I am not sure if this idea is the right one because I am stucking there... I think I have to use one of the convergence theorems because the given exercise follows in this chapter but I don't see how they used here in this context.

Do you have any ideas? Thank you for your help.



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