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:


 * $\displaystyle \int f \rd \mu < +\infty$
 * $\forall n \in \N: f_n \le f$

where $\le$ signifies a pointwise inequality.

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

Then:


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

where:


 * the integral sign denotes $\mu$-integration
 * the limit superior is taken in the extended real numbers $\overline \R$.

Also see

 * Fatou's Lemma for Integrals