Fatou's Lemma for Integrals/Positive Measurable Functions

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$, $f_n: X \to \overline \R$ be a sequence of positive measurable functions.

Let $\ds \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit inferior of the $f_n$.

Then:


 * $\ds \int \liminf_{n \mathop \to \infty} f_n \rd \mu \le \liminf_{n \mathop \to \infty} \int f_n \rd \mu$

where:


 * the integral sign denotes $\mu$-integration
 * the limit inferior is taken in the extended real numbers $\overline \R$.

Proof
For each $n \in \N$, define $g_n : X \to \overline \R$ by:


 * $\ds g_n = \inf_{k \mathop \ge n} f_k$

That is:


 * $\map {g_n} x = \inf \set {\map {f_k} x : k \ge n}$

for each $x \in X$.

For each $n \in \N$, we have that:


 * $\set {\map {f_k} x : k \ge n + 1} \subseteq \set {\map {f_k} x : k \ge n}$

From Infimum of Subset:


 * $\inf \set {\map {f_k} x : k \ge n + 1} \ge \inf \set {\map {f_k} x : k \ge n}$

That is:


 * $\map {g_{n + 1} } x \ge \map {g_n} x$

for each $n \in \N$.

From Pointwise Infimum of Measurable Functions is Measurable, we also have:


 * $g_n$ is $\Sigma$-measurable for each $n \in \N$.

We have:


 * $\ds \lim_{n \mathop \to \infty} g_n = \lim_{n \mathop \to \infty} \paren {\inf_{k \mathop \ge n} f_k}$

By the definition of limit inferior:


 * $\ds \lim_{n \mathop \to \infty} g_n = \liminf_{n \mathop \to \infty} f_n$

So:


 * $\sequence {g_n}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma$-measurable functions convergent to $\ds \liminf_{n \mathop \to \infty} f_n$.

We are therefore able to apply the monotone convergence theorem.

We then have::

Clearly we have:


 * $\ds \inf_{k \mathop \ge n} f_k \le f_n$

for any $n \in \N$.

So, by Integral of Integrable Function is Monotone:


 * $\ds \int \paren {\inf_{k \mathop \ge n} f_k} \rd \mu \le \int f_n \rd \mu$

and so:


 * $\ds \int \paren {\inf_{k \mathop \ge n} f_k} \rd \mu \le \inf_{k \mathop \ge n} \int f_k \rd \mu$

Then:

Also see

 * Reverse Fatou's Lemma/Positive Measurable Functions