Modes of Convergence: Difference between revisions
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*A sequence of measurable functions <math>\{f_n \}</math> converges to <math> f</math> pointwise almost everywhere if <math> f_n (x) \to f(x)</math> for almost every <math> x </math>, or <math> \mu( \{x: f(x) \neq \lim_{n \to \infty} f(x) \}) =0</math>. | *A sequence of measurable functions <math>\{f_n \}</math> converges to <math> f</math> pointwise almost everywhere if <math> f_n (x) \to f(x)</math> for almost every <math> x </math>, or <math> \mu( \{x: f(x) \neq \lim_{n \to \infty} f(x) \}) =0</math>. | ||
*A sequence of measurable functions <math>f_n</math> converges in <math>L^1</math> if <math>\int |f_n - f| \to 0.</math> | *A sequence of measurable functions <math>f_n</math> converges in <math>L^1</math> if <math>\int |f_n - f| \to 0.</math> | ||
check [[Convergence in Measure]] for convergence in measure. | |||
== 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>. | * <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>. | ||
* check [[Convergence in Measure]] for convergence in measure. | |||
Revision as of 18:55, 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 .
- check Convergence in Measure for convergence in measure.
