Prove lim as as $n \to \infty \int_{[-n,n]}{f} = \int_{\mathbb{R}}f$

Question

Prove that $$\lim_{n \to \infty} \int_{[-n,n]}{f} = \int_{\mathbb{R}}f.$$

We're given $f$ a nonnegative measurable function on $\mathbb{R}$.

So far I have:

Let $f_n = 1_{[-n,n]}f$ then $\{f_n\}$ is nonnegative and monotone and $ f \to f_n$ pointwise.

By MCT, $$\lim_{n \to \infty} \int_{[-n,n]}{f} = \lim_{n \to \infty} \int_{[-n,n]}{f_n} = \lim_{n \to \infty} \int_{\mathbb{R}}{f} = \int_{\mathbb{R}}{f}.$$

Is this right? I'm a little concerned about my last line

Answer 1

What you are doing is basically correct, but you are not writing it properly.

You have that $f_n\nearrow f$ (it is essential that the convergence is monotone).

Then $$ \lim_n\int_{[-n,n]}f=\lim_n\int_{\mathbb R} f_n=\int_{\mathbb R}\lim_n f_n =\int_{\mathbb R}f, $$ where the Monotone Convergence Theorem is used in the second equality.

Answer 2

You have the right idea. Be sure to demonstrate that $f_n$ is a sequence of measurable functions, and as @MartinArgerami says, demonstrate that you have a pointwise increasing sequence of functions.

I would describe that last line as follows:

$$\lim_{n\to\infty} \int_{[-n,n]} f dm = \lim_{n\to\infty} \int_{\mathbb{R}} f_n dm = \int_{\mathbb{R}} \lim_{n\to\infty} f_n dm = \int_{\mathbb{R}} f dm,$$ where the exchange of limits follows from the MCT.