Modes of Convergence: Difference between revisions

From Optimal Transport Wiki
Jump to navigation Jump to search
Line 8: Line 8:


== Relevant Properties <ref name="Craig, Katy">Craig, Katy. ''MATH 201A Lecture 18''. UC Santa Barbara, Fall 2020.</ref>==
== Relevant Properties <ref name="Craig, Katy">Craig, Katy. ''MATH 201A Lecture 18''. UC Santa Barbara, Fall 2020.</ref>==
* <math>f_n \to f</math> through    uniform Convergence <math>\to </math><math> f_n \to f</math> through  pointwise convergence  <math> \to </math> <math>f_n \to f</math> pointwise a.e. convergence
* <math>f_n \to f</math> through    uniform Convergence <math>\to </math><math> f_n \to f</math> through  pointwise convergence  <math> \to </math> <math>f_n \to f</math> pointwise a.e. convergence.
* <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>  in  <math>L^1</math>. To see example of why we need a dominating function, read [[Dominated Convergence Theorem]], particularly applications of the theorem.
* <math>f_n \to f</math> Pointwise a.e. convergence, equipped with dominating function, implies <math>f_n \to f</math>  in  <math>L^1</math>. To see example of why we need a dominating function, read [[Dominated Convergence Theorem]], particularly applications of the theorem.
* [[Convergence in Measure]] lists relationships between convergence in measure and other forms of convergence.
* [[Convergence in Measure]] lists relationships between convergence in measure and other forms of convergence.

Revision as of 19:00, 18 December 2020

Relevant Definitions[1]

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

check Convergence in Measure for convergence in measure.

Relevant Properties [2]

  • 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 in . To see example of why we need a dominating function, read Dominated Convergence Theorem, particularly applications of the theorem.
  • Convergence in Measure lists relationships between convergence in measure and other forms of convergence.
  1. Craig, Katy. MATH 201A Lecture 17. UC Santa Barbara, Fall 2020.
  2. Craig, Katy. MATH 201A Lecture 18. UC Santa Barbara, Fall 2020.