Reverse Fatou's Lemma/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.

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 $\overline \R$.

Proof

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Source of Name

This entry was named for Pierre Joseph Louis Fatou.

Also see

• Fatou's Lemma for Integrals

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $8$