Fatou's Lemma for Integrals/Positive Measurable Functions

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\left({f_n}\right)_{n \in \N} \in \mathcal M_{\overline \R}^+$, $f_n: X \to \overline \R$ be a sequence of positive measurable functions.

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


Then:

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

where:

the integral sign denotes $\mu$-integration; and
the right-hand side limit inferior is taken in the extended real numbers $\overline \R$.


Proof


The sequence $\displaystyle \inf_{m \mathop > n} f_m$ is a monotone increasing sequence.

Also:

$\displaystyle \inf_{m \mathop > n} f_m \le f_k$

for $k>n$.

We have then:

$\displaystyle \int \inf_{m \mathop > n} f_m \rd \mu \le \int f_k \rd \mu$

for $k > n$.

Thus:

$\displaystyle \int \inf_{m \mathop > n} f_m \rd \mu \le \inf_{m \mathop > n} \int f_m \rd \mu$

Now we take the limit $n \to \infty$.

Using the monotone convergence theorem we exchange integral and limit on the left side of the equation and obtain:

$\displaystyle \int \liminf_{m \mathop > n} f_m \rd \mu \le \liminf_{m \mathop > n} \int f_m \rd \mu$

$\blacksquare$


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Also see


Sources