Convergence in Measure: Difference between revisions

Revision as of 20:10, 10 December 2020

Let $(X,{\mathcal {M}},\mu )$ denote a measure space and let $f_{n},f:X\to \mathbb {R}$ for $n\in \mathbb {N}$ . The sequence $\{f_{n}\}_{n\in \mathbb {N} }$ converges to $f$ in measure if $\lim _{n\to \infty }\mu \left(\{x\in X:|f_{n}(x)-f(x)|\geq \epsilon \}\right)=0$ for any $\epsilon >0$ ^[1].

Further, the sequence $\{f_{n}\}_{n\in \mathbb {N} }$ is Cauchy in measure if for every $\epsilon >0$ $\mu (\{x\in X:|f_{n}(x)-f_{m}(x)|\geq \epsilon \})\to 0$ as $n,m\to \infty$ ^[1]

Properties

If $f_{n}\to f$ in measure and $g_{n}\to g$ in measure, then $f_{n}+g_{n}\to f+g$ in measure ^[2]

If $f_{n}\to f$ in measure and $g_{n}\to g$ in measure, then $f_{n}g_{n}\to fg$ in measure ^[2]

Relation to other types of Convergence

If $f_{n}\to f$ in $L^{1}(\mu )$ then $f_{n}\to f$ in measure ^[1]

If $f_{n}\to f$ in measure, then there exists a subsequence $\{f_{n_{k}}\}_{k\in \mathbb {N} }$ such that $f_{n_{k}}\to f$ almost everywhere.^[1]

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.4
↑ ^2.0 ^2.1 Katy Craig, Math 201a, Homework 8

[Folland-1] 1.0 ^1.1 ^1.2 ^1.3 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.4

[Craig1-2] 2.0 ^2.1 Katy Craig, Math 201a, Homework 8

[1]

[2]

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-Let <math>(X, \mathcal{M}, \mu)<math> denote a measure space and let <math>f_n, f : X \to \mathbb{R}<math> for <math>n \in \mathbb{N}<math>. The sequence <math>\{f_n\}_{n \in \mathbb{N}}<math> converges to <math>f<math> in measure if <math>\lim_{n \to \infty} \mu \left( \{x \in X : |f_n(x) - f(x)| \geq \epsilon \} \right) = 0<math> for any <math>\epsilon > 0<math>.
+Let <math>(X, \mathcal{M}, \mu)</math> denote a measure space and let <math>f_n, f : X \to \mathbb{R}</math> for <math>n \in \mathbb{N}</math>. The sequence <math>\{f_n\}_{n \in \mathbb{N}}</math> converges to <math>f</math> in measure if <math>\lim_{n \to \infty} \mu \left( \{x \in X : |f_n(x) - f(x)| \geq \epsilon \} \right) = 0</math> for any <math>\epsilon > 0</math> <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.4 </ref>.
+Further, the sequence <math> \{f_n\}_{n \in \mathbb{N}} </math> is Cauchy in measure if for every <math> \epsilon > 0 </math> <math> \mu(\{x \in X : |f_n(x) - f_m(x) | \geq \epsilon \}) \to 0 </math> as <math> n,m \to \infty </math> <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.4 </ref>
+==Properties==
+*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="Craig1">Katy Craig, ''Math 201a'', Homework 8 </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 <ref name="Craig1"></ref>
+==Relation to other types of Convergence==
+*If <math> f_n \to f</math> in <math> L^1(\mu) </math> then <math> f_n \to f </math> in measure <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.4 </ref>
+*If <math> f_n \to f </math> in measure, then there exists a subsequence <math> \{f_{n_k}\}_{k \in \mathbb{N}} </math> such that <math> f_{n_k} \to f </math> almost everywhere.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §2.4 </ref>
+==References==

Convergence in Measure: Difference between revisions

Revision as of 20:10, 10 December 2020

Properties

Relation to other types of Convergence

References

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