Fatou's Lemma: Difference between revisions

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==Proof==
==Proof==
For <math> \forall n \in \mathbb{N} </math>
For any <math> n \in \mathbb{N} </math>, let <math> g_n=\inf_{k\geq n} f_k </math>.
 
By definition, <math> \liminf_{n\rightarrow +\infty} f_n= \lim_{n\rightarrow +\infty} (inf_{k\geq n}f_k)=\lim_{n\rightarrow +\infty} g_n</math>.
And <math> g_n\leq g_{n+1}, \forall n \in \mathbb{N} </math>, so by Monotone Convergence Theorem,
<math> \lim_{n\rightarrow +\infty} \int g_n=\int \lim_{n\rightarrow +\infty} g_n = \int \liminf_{n\rightarrow +\infty} f_n</math>.
 
Furthermore, by definition we have <math> g_n\leq f_n </math>, then <math> \int g_n\leq \int f_n </math>.
 
Since <math> \lim_{n\rightarrow +\infty} </math> exists, taking <math> \liminf_{n\rightarrow +\infty} </math> of both sides:
<math> \int \liminf_{n\rightarrow +\infty} f_n=\lim_{n\rightarrow +\infty}= \liminf_{n\rightarrow +\infty} \int g_n \leq \liminf_{n\rightarrow +\infty} \int f_n</math>.


==References==
==References==

Revision as of 05:28, 6 December 2020

Statement

Suppose is a sequence of non-negative measurable functions, . Then: . [1]

Proof

For any , let .

By definition, . And , so by Monotone Convergence Theorem, .

Furthermore, by definition we have , then .

Since exists, taking of both sides: .

References

  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2