Fatou's Lemma for Integrals/Integrable Functions
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Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.
Let $\left({f_n}\right)_{n \in \N} \in \mathcal{L}^1$, $f_n: X \to \R$ be a sequence of integrable functions.
Let $\displaystyle \liminf_{n \to \infty} f_n: X \to \overline{\R}$ be the pointwise limit inferior of the $f_n$.
Suppose that there exists an integrable $f: X \to \R$ such that for all $n \in \N$, $f \le f_n$ pointwise.
Then:
- $\displaystyle \int \liminf_{n \to \infty} f_n \, \mathrm d\mu \le \liminf_{n \to \infty} \int f_n \, \mathrm d\mu$
where:
- the integral sign denotes $\mu$-integration; and
- the right-hand side limit inferior is taken in the extended real numbers $\overline{\R}$.
Proof
Source of Name
This entry was named for Pierre Joseph Louis Fatou.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 10$: Problem $10.8 \ \text{(i)}$