Fatou's Lemma for Integrals
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.
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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 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 $\ds \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$ 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:
- $\ds \int \liminf_{n \mathop \to \infty} f_n \rd \mu \le \liminf_{n \mathop \to \infty} \int f_n \rd \mu$
where:
- the integral sign denotes $\mu$-integration
- the 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.