Dominated Convergence Theorem: Difference between revisions
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==Statement and proof of Theorem== | ==Statement and proof of Theorem== | ||
====Statement==== | ====Statement==== | ||
Suppose <math>\{f_n\}</math> is a sequence of [[Measurable function | measurable functions]] such that <math>f_n(x) \to f(x)</math> a.e. x, as n goes to infinity. If <math>|f_n(x)|\leq g(x) </math>, where g is integrable, then | Suppose <math>\{f_n\}</math> is a sequence of [[Measurable function | measurable functions]] such that <math>f_n(x) \to f(x)</math> a.e. x, as n goes to infinity. If <math>|f_n(x)|\leq g(x) </math>, where g is integrable, then <math>f \in \mathrb{L^1}</math> and | ||
and | |||
:<math>\int f_n \to \int f</math> | :<math>\int f_n \to \int f</math> | ||
Revision as of 04:09, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in \mathrb{L^1}} and
