# Excercise on Lebesgue-Integral

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.