Modes of Convergence: Difference between revisions
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== Relevant Definitions<ref name="Craig">Craig, Katy. ''MATH 201A Lecture 17''. UC Santa Barbara, Fall 2020.</ref>== | == Relevant Definitions<ref name="Craig">Craig, Katy. ''MATH 201A Lecture 17''. UC Santa Barbara, Fall 2020.</ref>== | ||
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 <math>\{x \in X: p(x) \text{ doesn't hold }\}</math> has measure zero. | 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 <math>\{x \in X: p(x) \text{ doesn't hold }\}</math> has measure zero. | ||
* A sequence of functions <math>f_n</math> converges pointwise if <math>f_n(x) \to f(x) </math> for all <math>x \in X </math>. | * A sequence of measurable functions <math>\{f_n \}</math> converges pointwise if <math>f_n(x) \to f(x) </math> for all <math>x \in X </math>. | ||
* A sequence of functions <math>f_n</math> converges uniformly if <math>\sup_{x \in X} |f_n(x) - f(x)| \to 0 </math>. | * A sequence of measureable functions <math>\{f_n \}</math> converges uniformly if <math>\sup_{x \in X} |f_n(x) - f(x)| \to 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 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> | ||
Revision as of 19:02, 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 measurable functions converges pointwise if for all .
- A sequence of measureable 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 cnvergence implies through pointwise convergence pointwise a.e. convergence.
- through convergence implies 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.
