Dominated Convergence Theorem: Difference between revisions

Revision as of 04:08, 18 December 2020

In measure theory, the dominated convergence theorem is a cornerstone of Lebesgue integration. It can be viewed as a culmination of all efforts, and is a general statement about the interplay between limits and integrals.

Statement and proof of Theorem

Statement

Suppose $\{f_{n}\}$ is a sequence of measurable functions such that $f_{n}(x)\to f(x)$ a.e. x, as n goes to infinity. If $|f_{n}(x)|\leq g(x)$ , where g is integrable, then

\int |f_{n}-f|\to 0

and consequently

\int f_{n}\to \int f

@@ Line 6: / Line 6: @@
 and consequently
 :<math>\int f_n \to \int f</math>
-====Lemma====
-Suppose <math>f</math> is integrable in <math>R</math>. Then for every <math>\epsilon>0</math>, there exist a set of finite measure B such that
-:<math>\int_{B^c} |f|<\epsilon</math>
-*Proof of lemma: By replacing <math>f</math> with <math>|f|</math> we may assume without loss of generality that <math>f\geq 0</math>.Let <math>B_N</math> denote the ball of radius N centered at origin, and note that if <math>f_N(x)=f(x)\mathbf{1}_{B_N}(x)</math>, then <math>f_N\geq0 is measurable</math>, <math>f_N(x)\leq f_{N+1}(x)</math>, and <math>f_N(x)\to f(x)</math>. By the monotone convergenc theorem, we must have

Dominated Convergence Theorem: Difference between revisions

Revision as of 04:08, 18 December 2020

Statement and proof of Theorem

Statement

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