Prove that $\lim_{n\rightarrow \infty} \int_{[-n,n]} f\,d\lambda= \int f\,d\lambda.$

lebesgue-integrallebesgue-measuremeasure-theoryreal-analysis

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 $$\lim_{n\rightarrow \infty}
\int_{[-n,n]} f\,d\lambda= \int f\,d\lambda.$$

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 $E\in \mathcal{S}$, and an $\mathcal{S}$-measurable function $f$, $$\int_E f\,d\mu=\int f\chi_{E}\,d\mu$$ 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 $n\in\mathbf{N}$?

  3. If (2) is true, then I can define $f_n=f\chi_{E_n}$ and $$\lim_{n\rightarrow\infty}f_n=f\chi_E.$$ Then I think I would have my answer by the Dominated Convergence Theorem.

Am I on the right track here?

Best Answer

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.

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