Reverse 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.
Suppose that there exists a positive measurable function $f: X \to \overline \R$ such that:
- $\ds \int f \rd \mu < +\infty$
- $\forall n \in \N: f_n \le f$
where $\le$ signifies a pointwise inequality.
Let $\ds \limsup_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit superior of the $f_n$.
Then:
- $\ds \limsup_{n \mathop \to \infty} \int f_n \rd \mu \le \int \limsup_{n \mathop \to \infty} f_n \rd \mu$
where:
- the integral sign denotes $\mu$-integration
- the left hand side limit superior 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.
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Let $\ds \limsup_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit superior of the $f_n$.
Suppose that there exists an $\mu$-integrable $f: X \to \R$ such that for all $n \in \N$, $f_n \le f$ pointwise.
Then:
- $\ds \limsup_{n \mathop \to \infty} \int f_n \rd \mu \le \int \limsup_{n \mathop \to \infty} f_n \rd \mu$
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.