[Math] Real Analysis, 2.18 (Fatou’s Lemma) Integration of Nonnegative functions

Question

2.18 Fatou's Lemma – If $\{f_n\}$ is any sequence in $L^+$, then $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) \leq \lim_{n\rightarrow \infty}\inf\int f_n$$

Attempted proof – We know that $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) = \int \sup_{k\geq 1}\left(\inf_{n\geq k}f_n\right) = \int \lim_{k\rightarrow \infty}\inf_{n\geq k}f_n$$ Then by the Monotone Convergence theorem $$\int \lim_{k\rightarrow \infty}\inf_{n\geq k}f_n = \lim_{k\rightarrow \infty}\int \inf_{n\geq k}f_n$$ Note that the Monotone Convergence theorem can be applied because $$\inf_{n\geq k} f_n \leq \inf_{n\geq k+1} f_n$$

We see that $\inf_{n\geq k}f_n \leq f_k$ for all $n\geq k$. So,
\begin{align*}
\int \inf_{n\geq k}f_n &\leq \int f_k \ \forall n\geq k\\
&\leq\inf_{n\geq k}\int f_n\\
&\leq \lim_{k\rightarrow \infty}\inf_{n\geq k}\int f_n
\end{align*}

I may have some indexing mistakes but I think this is a sufficient proof. Any suggestions is greatly appreciated.

Answer 1

Your proof is essentially correct. Just some minor adjustments are required.

2.18 Fatou's Lemma - If $\{f_n\}$ is any sequence in $L^+$, then $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) \leq \lim_{n\rightarrow \infty}\inf\int f_n$$

Proof - We know that $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) = \int \sup_{k\geq 1}\left(\inf_{n\geq k}f_n\right) = \int \lim_{k\rightarrow \infty}\inf_{n\geq k}f_n$$ Then by the Monotone Convergence theorem $$\int \lim_{k\rightarrow \infty}\inf_{n\geq k}f_n = \lim_{k\rightarrow \infty}\int \inf_{n\geq k}f_n \tag{1}$$ Note that the Monotone Convergence theorem can be applied because $$\inf_{n\geq k} f_n \leq \inf_{n\geq k+1} f_n$$ in other words, $\{\inf_{n\geq k} f_n\}_k$ is a non-decreasing sequence of non-negative functions.

We see that $\inf_{n\geq k}f_n \leq f_n$ for all $n \geq k$. So, \begin{align*} \int \inf_{n\geq k}f_n &\leq \int f_n \ \forall n\geq k\\ &\leq\inf_{n\geq k}\int f_n \end{align*}

Since $\inf_{n\geq k}\int f_n$ is a non-decreasing sequence of (extended) real numbers, there is $\lim_{k \to +\infty}\inf_{n\geq k}\int f_n$ and we get, from $(1)$, $$\int \lim_{k\rightarrow \infty}\inf_{n\geq k}f_n = \lim_{k\rightarrow \infty}\int \inf_{n\geq k}f_n \leq \lim_{k \to +\infty}\inf_{n\geq k}\int f_n$$