Royden's Lebesgue Integration 4.4 #34

Question

Let $f$ be a nonegative measurable function on $\mathbb{R}$. Show that

\begin{equation} \boxed{\lim_{n \to \infty} \int_{-n}^{n} f=\int_{\mathbb{R}} f} \end{equation}

Of course $$\forall n, \int_{-n}^{n} f \leq\int_{\mathbb{R}} f$$

To show the reverse, I feel that I need to use Fatou's lemma; but can't seem to get the right argument. This problem is found in Royden's Real Analysis 4th edition, p. 90, #34.

Answer 1

The sequence of integrals is not your main concern. The sequence of functions $f_n(x)=f(x)\cdot \mathbb{1}_{(-n, n)}$ is your main concern.

Answer 2

You may apply the dominated convergence theorem with $$ \int_{-n}^{n} f=\int_{\mathbb{R}}f(x)\cdot \mathbb{1}_{(-n, n)} \leq\int_{\mathbb{R}} f $$ and you may conclude easily.

Answer 3

Olivier Oloa's answer says you should use the dominated convergence theorem.

Giuseppe Negro's answer says you should consider this sequence of functions: $$ f_n(x)=f(x)\cdot \mathbb{1}_{(-n, n)}. \tag 1 $$

I would have considered the sequence $(1)$ and applied the monotone convergence theorem.

However, the question of whether that is what you ought to do depends on the context. Should you just cite and apply the theorem, or should you prove this particular instance of it from scratch? If you haven't yet seen the theorem at that point in the book, maybe the latter. If you have seen it and the exercise is phrased in a certain way, then also the latter. If that section of the book was devoted to proving the monotone convergence theorem, then probably the former.