Dominated Convergence Theorem: Difference between revisions
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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 | 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> | :<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\leq 0</math> |
Revision as of 03:58, 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 is a sequence of measurable functions such that a.e. x, as n goes to infinity. If , where g is integrable, then
and consequently
Lemma
Suppose is integrable in . Then for every , there exist a set of finite measure B such that
- Proof of lemma: By replacing with we may assume without loss of generality that