Two inequalities about using Fatou Lemma

1. We can show a stronger result, namely, that $\int_{\mathbb R}\left\lvert f_n-f\right\rvert\to 0$, by applying Fatou's lemma to $g_n:= \left\lvert f_n-f\right\rvert-f+f_n$.

2. If $\left(a_n\right)_{n\geqslant 1}$ is a sequence of real number such that $a_n\leqslant t$ for all $n$, then $\limsup_{n\to +\infty}a_n\leqslant t$, because for all $N$, $s_N\sup_{n\geqslant N}a_n\leqslant t$ and $\limsup_{n\to +\infty}a_n=\lim_{N\to +\infty}s_N$.

Clearly $|f_n - f| \le f_n + f$ so define $g_n = f_n + f - |f_n - f| \ge 0$. We have $g_n \xrightarrow{n\to\infty} 2f$ pointwise.

Fatou's lemma applied on $g_n$ gives

\begin{align} 2\int_{\mathbb{R}} f &\le \liminf_{n\to\infty} \int_\mathbb{R} g_n \\ &= \liminf_{n\to\infty} \int_\mathbb{R} f_n + \liminf_{n\to\infty} \int_\mathbb{R} f + \liminf_{n\to\infty} \left(-\int_\mathbb{R}|f_n - f|\right) \\ &= 2\int_{\mathbb{R}} f - \limsup_{n\to\infty} \int_\mathbb{R}|f_n - f| \end{align}

so $\limsup_{n\to\infty} \int_\mathbb{R}|f_n - f| = 0$, which implies $\int_\mathbb{R}|f_n - f| \xrightarrow{n\to\infty} 0$.

Now for any $E \subseteq \mathbb{R}$ measurable

$$\left|\int_E f_n - \int_E f\right| = \left|\int_E (f_n - f)\right| \le \int_E |f_n - f| \le \int_\mathbb{R} |f_n - f| \xrightarrow{n\to\infty} 0$$

so $\int_E f_n \xrightarrow{n\to\infty} \int_E f$.

Good question. Here is another possible approach.

You already get $\int_{\mathbb R} f = \int_{\mathbb R \setminus E} f + \int_{E} f$ and $\int_{\mathbb R \setminus E} f \le \liminf_{n\to {\infty}}\biggr(\int_{\mathbb R \setminus E} f_n \biggr)$. So we have

$$ \lim_{n\to {\infty}}\int_{\mathbb R} f_n =\int_{\mathbb R} f \leq\liminf_{n\to {\infty}}\biggr(\int_{\mathbb R \setminus E} f_n \biggr)+ \int_{E} f. $$

Notice that for real sequences $a_n$, $\liminf_{n\to \infty}a_n = -\limsup_{n\to \infty}(-a_n)$ (see the "property" section in wiki), and $\lim_{n\to {\infty}}\int_{\mathbb R} f_n = \limsup_{n\to \infty}\int_{\mathbb R} f_n$. Thus the inequality above can be rewritten as $$ \limsup_{n\to {\infty}}\int_{\mathbb R} f_n \leq-\limsup_{n\to {\infty}}\biggr(-\int_{\mathbb R \setminus E} f_n \biggr)+ \int_{E} f, $$ or equivalently, $$ \limsup_{n\to {\infty}}\int_{\mathbb R} f_n +\limsup_{n\to {\infty}}\biggr(-\int_{\mathbb R \setminus E} f_n \biggr)\leq \int_{E} f, $$

Next we will use another property for limit superior, which says that for two sequences $a_n$ and $b_n$, we have

$$ \limsup_{n\to \infty}(a_n+b_n)\leq \limsup_{n\to \infty}a_n + \limsup_{n\to \infty}b_n. $$

Therefore we have $$ \limsup_{n\to {\infty}}\biggr(\int_{\mathbb R} f_n-\int_{\mathbb R \setminus E} f_n \biggr)\leq \limsup_{n\to {\infty}}\int_{\mathbb R} f_n +\limsup_{n\to {\infty}}\biggr(-\int_{\mathbb R \setminus E} f_n \biggr)\leq \int_{E} f, $$

that is, $$\int_{E} f\geq \limsup_{n\to {\infty}}\int_{ E} f_n.$$

For the last step, by Fatou's lemma we have $\int_{E} f\leq \liminf_{n\to {\infty}}\int_{ E} f_n$, thus

$$ \lim_{n\to {\infty}}\int_{ E} f_n = \liminf_{n\to {\infty}}\int_{ E} f_n = \limsup_{n\to {\infty}}\int_{ E} f_n = \int_{E} f. $$