Show that $\int_{\mathbb{R}}f = \lim_{n \to \infty} \int_{\mathbb{R}}f_n$ given specific assumptions

Question

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem:

If $f, f_n: \mathbb{R} \to [0, +\infty)$ measurable, $f_n \to f$ pointwise on $\mathbb{R}$ and $f_n \leq f, \forall n \in \mathbb{N}$, show that $\int_{\mathbb{R}}f = \lim_{n \to \infty} \int_{\mathbb{R}}f_n$.

The assumptions are (I always rewrite the problem to see if I understand it correctly):

1. $f$ and $f_n$ are nonnegative measurable functions.
2. $\lim_{n \to \infty}f_n = f$ (pointwise limit).
3. $f_n \leq f$ for all $n$.

If I could commute the limit and the integral then I would be done (but it is never that simple, unfortunately).

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This problem looks like an application of Fatou's lemma, i.e.

If $f_n$ is a sequence of nonnegative functions, then $$\int \Big(\liminf_{n \to \infty} \ f_n \Big) \leq \liminf_{n \to \infty}\int f_n$$

but I'm having difficulities applying it to this specific problem.

Answer 1

Your assumption about Fatou's lemma is correct. Note that, using your assumptions, the left hand sides equals $\int f$. Now, on the other hand, by monotonicity of the integral, $\int f_n\le \int f$ for every $n$. Can you complete the reasoning from here?