Modes of Convergence: Difference between revisions

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(Created page with "== Relevant Definitions== Denote our measure space as <math> (X, \mathcal{M}, \mu) </math>. Note that a property p(x) holds for almost every <math>x \in X</math> if the set <m...")
 
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* <math>f_n \to f</math> through  <math> L^1</math> convergence <math>\to  </math> <math>f_n \to f</math>  through pointwise a.e convergence up to a subsequence
* <math>f_n \to f</math> through  <math> L^1</math> convergence <math>\to  </math> <math>f_n \to f</math>  through pointwise a.e convergence up to a subsequence
* <math>f_n \to f</math> Pointwise a.e. convergence equipped with dominating function implies <math>f_n \to f</math>  <math>L^1</math> convergence
* <math>f_n \to f</math> Pointwise a.e. convergence equipped with dominating function implies <math>f_n \to f</math>  <math>L^1</math> convergence
<ref name="Craig, Katy">Craig, Katy. ''MATH 201A Lecture 18''. UC Santa Barbara, Fall 2020.</ref>
<ref name="Craig, Katy">Craig, Katy. ''MATH 201A Lecture 19''. UC Santa Barbara, Fall 2020.</ref>

Revision as of 07:41, 18 December 2020

Relevant Definitions

Denote our measure space as . Note that a property p(x) holds for almost every if the set has measure zero.

  • A sequence of functions converges pointwise if for all
  • A sequence of functions converges uniformly if ,
  • A sequence of measurable functions converges to pointwise almost everywhere if for almost every , or .
  • A sequence of measurable functions converges in if

Relevant Properties

  • through uniform Convergence through pointwise convergence pointwise a.e. convergence
  • through convergence through pointwise a.e convergence up to a subsequence
  • Pointwise a.e. convergence equipped with dominating function implies convergence

[1] [1]

  1. 1.0 1.1 Craig, Katy. MATH 201A Lecture 18. UC Santa Barbara, Fall 2020. Cite error: Invalid <ref> tag; name "Craig, Katy" defined multiple times with different content