Fatou's Lemma

[[concept]]

Topics

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Theorem (Fatou's Lemma)

Let $\{f_n{\}}_n$ be a sequence in $L^{+}(E)$. Then ${\int }_E{\lim }_{n\to \infty }\inf f_n(x)dx\le {\lim }_{n\to \infty }\inf \int f_n(x)dx$

Recall ${\lim }_{n\to \infty }\inf a_n={\sup }_{n\ge 1}[{\inf }_{k\ge n}{a}_k]$ And the ${\lim }_{n\to \infty }\inf$ function is defined pointwise

Proof

We have

\lim_{ n \to \infty } \inf f_{n}(x) &= \sup_{n \geq 1} [\inf_{k \geq n} f_{k}(x)] \\ &= \lim_{ n \to \infty } [\inf_{k \geq n} f_{k}(x)] \end{align}$$ And since $\inf_{k \geq 1} f_{k}(x) \leq \inf_{k \geq 2} f_{k} \leq\dots$ by the [[Concept Wiki/Monotone Convergence Theorem\|MCT]] we have $$\int _{E} \lim_{ n \to \infty } \inf f_{n} = \lim_{ n \to \infty } \int _{E} \inf_{k \geq n} f_{k}$$ For all $j \geq n$ and all $x \in E$, we have $$\begin{align} \inf_{k \geq n} f_{k}(x) &\leq f_{j} (x) \\ \implies \int _{E} \inf_{k \geq n} f_{k} &\leq \int _{E} f_{j} \\ \implies \int _{E} \lim_{ n \to \infty } \inf f_{n} &= \lim_{ n \to \infty } \int _{E} (\inf_{k \geq n} f_{k}) \\ &\leq \lim_{ n \to \infty } \left[\inf_{j \geq n} \int _{E} f_{j}\right] \\ &= \lim_{ n \to \infty } \inf \int _{E}f_{n} \end{align}$$ So we have "swapped the integral and inf" and we can plug this into the MCT to get the desired result. $$\tag*{$\blacksquare$}$$

References

References

See Also

Mentions

Mentions

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