# Help with the Lebesgue Integration for nonnegative, measurable functions.

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.

• Does $E_n\subset E_{n+1}$? Commented Oct 23, 2021 at 2:03
• draw pictures is my suggestion Commented Oct 23, 2021 at 2:16
• $E_n \nearrow E = \{x\in \mathbb{R} : f(x) > 0\}$ and $[-n, n] \nearrow \mathbb{R}$. Now apply monotone convergence theorem and note that since $f = 0$ on $E^c$, $\int_{\mathbb{R}}f\,dm = \int_{E}f\,dm$. Commented Oct 23, 2021 at 22:13

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 $$\lim_{n\to \infty} \int_{[-n,n]}f dm=\lim_{n\to \infty} \int_{\mathbb R}f\chi_{[-n,n]} dm=\int_{\mathbb R}\lim_{n\to \infty}\left(f\chi_{[-n,n]}\right) dm=\int_\mathbb R f dm\ .$$
• Thank you, that's helpful! I am still having trouble with changing the bounds of the integral. How do we go from $\mathbb{R}$ to $[-n,n]$ to $E_n$? I know we have independence of representation but I am having trouble applying it here.