Fatou's Lemma: Difference between revisions

Latest revision as of 02:33, 12 December 2020

Statement

Suppose $\{f_{n}\}$ is a sequence of non-negative measurable functions, $f_{n}:X\to [0,+\infty ]$ . Then:

$\int \liminf _{n\rightarrow +\infty }f_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$ . ^[1]

Proof^[2]

Define $g_{n}:=\inf _{k\geq n}f_{k}$ for all $n\in \mathbb {N}$ .

By definition, $\liminf _{n\rightarrow +\infty }f_{n}=\lim _{n\rightarrow +\infty }(\inf _{k\geq n}f_{k})=\lim _{n\rightarrow +\infty }g_{n}$ and $g_{n}\leq g_{n+1},\forall n\in \mathbb {N}$ , so by Monotone Convergence Theorem,

$\lim _{n\rightarrow +\infty }\int g_{n}=\int \lim _{n\rightarrow +\infty }g_{n}=\int \liminf _{n\rightarrow +\infty }f_{n}$ .

Furthermore, by definition we have $g_{n}\leq f_{n},\forall n\in \mathbb {N}$ , implying that $\int g_{n}\leq \int f_{n}$ .

Since $\lim _{n\rightarrow +\infty }\int g_{n}$ exists, taking $\liminf _{n\rightarrow +\infty }$ of both sides yields:

$\int \liminf _{n\rightarrow +\infty }f_{n}=\lim _{n\rightarrow +\infty }\int g_{n}=\liminf _{n\rightarrow +\infty }\int g_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$ .

References

↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2
↑ Craig, Katy. MATH 201A Lecture 14. UC Santa Barbara, Fall 2020.

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

[2] Craig, Katy. MATH 201A Lecture 14. UC Santa Barbara, Fall 2020.

[1]

[2]

@@ Line 2: / Line 2: @@
 Suppose <math>\{f_n\}</math> is a sequence of non-negative measurable functions, <math> f_n: X \to [0,+\infty]</math>.
 Then:
 <math> \int \liminf_{n\rightarrow +\infty} f_n \leq \liminf_{n\rightarrow +\infty}\int f_n </math>. <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.2 </ref>
-==Proof==
+==Proof<ref>Craig, Katy. ''MATH 201A Lecture 14''. UC Santa Barbara, Fall 2020.</ref>==
-For any <math> n\in\mathbb{N} <math>
+Define <math> g_n := \inf_{k\geq n} f_k </math> for all <math> n \in \mathbb{N} </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,  \forall n \in \mathbb{N}</math>, implying that <math> \int g_n\leq \int f_n </math>.
+Since <math> \lim_{n\rightarrow +\infty} \int g_n </math> exists, taking <math> \liminf_{n\rightarrow +\infty} </math> of both sides yields:
+<math> \int \liminf_{n\rightarrow +\infty} f_n=\lim_{n\rightarrow +\infty} \int g_n = \liminf_{n\rightarrow +\infty} \int g_n \leq \liminf_{n\rightarrow +\infty} \int f_n</math>.
 ==References==

Fatou's Lemma: Difference between revisions

Latest revision as of 02:33, 12 December 2020

Statement

Proof[2]

References

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Proof^[2]