Fatou's Lemma for Integrals

From ProofWiki
Jump to navigation Jump to search

This proof is about Fatou's Lemma in the context of Integral of Positive Measurable Function. For other uses, see Fatou's Lemma.

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.


Positive Measurable Functions

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


Then:

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

where:

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


Integrable Functions

Let $\sequence {f_n}_{n mathop \in \N} \in \LL^1$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions.

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

Suppose that there exists an $\mu$-integrable $f: X \to \R$ such that for all $n \in \N$, $f \le f_n$ pointwise.


Then:

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

where:

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


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Also see