Fatou's Lemma

Theorem (Fatou's Lemma)

Let ${f_{n}}_{n}$ be a sequence in $L^{+} (E)$ . Then

\int_{E} lim_{n \to \infty} inf f_{n} (x) d x \leq lim_{n \to \infty} inf \int f_{n} (x) d x

Recall

$lim_{n \to \infty} inf a_{n} = sup_{n \geq 1} [inf_{k \geq n} a_{k}]$

And the $lim_{n \to \infty} inf$ function is defined pointwise

Proof

We have

\begin{aligned} lim_{n \to \infty} inf f_{n} (x) & = sup_{n \geq 1} [inf_{k \geq n} f_{k} (x)] \\ = lim_{n \to \infty} [inf_{k \geq n} f_{k} (x)] \end{aligned}

And since $inf_{k \geq 1} f_{k} (x) \leq inf_{k \geq 2} f_{k} \leq \dots$ by the MCT we have

\int_{E} lim_{n \to \infty} inf f_{n} = lim_{n \to \infty} \int_{E} inf_{k \geq n} f_{k}

For all $j \geq n$ and all $x \in E$ , we have

\begin{aligned} inf_{k \geq n} f_{k} (x) & \leq f_{j} (x) \\ ⟹ \int_{E} inf_{k \geq n} f_{k} & \leq \int_{E} f_{j} \\ ⟹ \int_{E} lim_{n \to \infty} inf f_{n} & = lim_{n \to \infty} \int_{E} (inf_{k \geq n} f_{k}) \\ \leq lim_{n \to \infty} [inf_{j \geq n} \int_{E} f_{j}] \\ = lim_{n \to \infty} inf \int_{E} f_{n} \end{aligned}

So we have "swapped the integral and inf" and we can plug this into the MCT to get the desired result.

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File	Last Modified
Functional Analysis Lecture 11	2025-07-15
Functional Analysis Lecture 13	2025-07-08

References