Cfrgtkky

I am working on the following exercise:

Let $lambda$ denote Lebesgue measure on $mathbf{R}$. Suppose
$f:mathbf{R}rightarrow mathbf{R}$ is a Borel measurable function
such that $int|f|<infty$. Prove that $$lim_{nrightarrow infty}
int_{[-n,n]} f,dlambda= int f,dlambda.$$

Now, I have some ideas as to things that could help me here, but I can't really put any of it together. Here is what I have so far:

1. I know that, given a measure space $(X,mathcal{S},mu)$, a set $Ein mathcal{S}$, and an $mathcal{S}$-measurable function $f$, $$int_E f,dmu=int fchi_{E},dmu$$ if the RHS is defined. In this case, I believe it is, since $int |f|<infty$.




2. Then this question got me thinking: what if I write $[-n,n]$ as the limit of an increasing sequence of sets? That is, can I write $E=[-n,n]=bigcup_{n=1}^infty E_n$ where $E_n=[-n,n]$ for $ninmathbf{N}$?




3. If (2) is true, then I can define $f_n=fchi_{E_n}$ and $$lim_{nrightarrowinfty}f_n=fchi_E.$$ Then I think I would have my answer by the Dominated Convergence Theorem.

Am I on the right track here?

Your proof is fine as-is.

For an alternative proof, you could apply the monotone convergence theorem to the sequences $f^+ chi_{E_n}$ and $f^{-} chi_{E_n}$ separately.