Dominated Convergence Theorem: Difference between revisions
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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. | 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 and proof of Theorem== | ||
*Statement: Suppose <math>\{f_n\}</math> is a sequence of measurable functions such that f_n(x) \to f(x) | *Statement: Suppose <math>\{f_n\}</math> is a sequence of measurable functions such that <math>f_n(x) \to f(x)</math> a.e. x, as n goes to infinity. | ||
Revision as of 03:43, 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.
