Fatou's lemma

Remarks

Proof idea: let $g_k = \inf_{n \ge k} f_n$. The functions $g_k$ increase to $\liminf_n f_n$, and $\int g_k \le \int f_n$ for every $n \ge k$. The monotone convergence theorem gives $\int \liminf_n f_n = \lim_k \int g_k \le \liminf_n \int f_n$.

The inequality can be strict. For $f_n = n \mathbf{1}_{(0, 1/n)}$ on $\R$ with Lebesgue measure, $f_n \to 0$ pointwise but $\int f_n = 1$ for all $n$.

Non-negativity is needed. For $f_n = -\mathbf{1}_{[n, n+1]}$ on $\R$, the left side is $0$ and the right side is $-1$.

Going deeper

For a student

Fatou's lemma is the step between the monotone convergence theorem and the dominated convergence theorem. It is often used to show that a limit function is integrable: if $\int f_n \le C$ for all $n$, then $\int \liminf f_n \le C$. It also shows that the $L^p$ norm is lower semicontinuous under pointwise convergence almost everywhere.