Help with the Lebesgue Integration for nonnegative, measurable functions.

Question

Let $(X, \mathcal{M}, \mu)$ be a measueable space, and let $f: \mathbb{R}\rightarrow \mathbb{R}$ be nonnegative $f(x)\geq0$ and measurable. Let $E_n=\{x\in \mathbb{R} : f(x)\geq \frac{1}{n}\}$. Show the following result

$$\int_{\mathbb{R}}fdm=\lim_{n \to \infty} \int_{[-n,n]}fdm=\lim_{n\to \infty} \int_{E_n}fdm.$$

I was thinking maybe we can assume $f\chi_{E_n}\leq f$, monotone increasing sequence of nonnegative measurable functions, and then apply monotone convergence theorem, but I am not sure that we can find such increasing sequence, as $\frac{1}{n}$ is decreasing.

Answer 1

Both sequences $$f\chi_{E_n},f\chi_{[-n,n]}$$ are increasing because $f$ is non-negative, $E_n\subset E_{n+1}$, and $[-n,n]\subset [-(n+1),n+1]$. Now using monotone convergence theorem we have $$\lim_{n\to \infty} \int_{E_n}f dm=\lim_{n\to \infty} \int_{\mathbb R}f\chi_{E_n} dm=\int_{\mathbb R}\lim_{n\to \infty}\left(f\chi_{E_n}\right) dm=\int_\mathbb R f dm$$ and similarly $$\lim_{n\to \infty} \int_{[-n,n]}f dm=\lim_{n\to \infty} \int_{\mathbb R}f\chi_{[-n,n]} dm=\int_{\mathbb R}\lim_{n\to \infty}\left(f\chi_{[-n,n]}\right) dm=\int_\mathbb R f dm\ .$$