Convergence in Measure: Difference between revisions
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*If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_n+g_n \to f+g </math> in measure.<ref name="Craig">Craig, Katy. ''MATH 201A HW 8''. UC Santa Barbara, Fall 2020.</ref> | *If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_n+g_n \to f+g </math> in measure.<ref name="Craig">Craig, Katy. ''MATH 201A HW 8''. UC Santa Barbara, Fall 2020.</ref> | ||
*If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_ng_n \to fg </math> in measure. | *If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_ng_n \to fg </math> in measure if <math> \mu(X) < \infty </math>. | ||
*If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_ng_n \to fg </math> in measure if this is a finite measure space. <ref name="Craig">Craig, Katy. ''MATH 201A HW 8''. UC Santa Barbara, Fall 2020.</ref> | *If <math> f_n \to f </math> in measure and <math> g_n \to g </math> in measure, then <math> f_ng_n \to fg </math> in measure if this is a finite measure space. <ref name="Craig">Craig, Katy. ''MATH 201A HW 8''. UC Santa Barbara, Fall 2020.</ref> |
Revision as of 23:20, 16 December 2020
Let denote a measure space and let for . The sequence converges to in measure if for any . Furthermore, the sequence is Cauchy in measure if for every as [1]
Properties
- If in measure and in measure, then in measure.[2]
- If in measure and in measure, then in measure if .
- If in measure and in measure, then in measure if this is a finite measure space. [2]
Relation to other types of Convergence
- If in then in measure [1]
- If in measure, then there exists a subsequence such that almost everywhere.[1]