Convergence in Measure: Difference between revisions
Jump to navigation
Jump to search
Blainequack (talk | contribs) (Created page with "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...") |
Blainequack (talk | contribs) No edit summary |
||
| Line 1: | Line 1: | ||
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>. | ||
Revision as of 19:12, 10 December 2020
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>.