Pointwise a.e. Convergence: Difference between revisions
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== Definition == | |||
Denote our measure space as <math> (X, \mathcal{M}, \mu) </math>. | |||
Define the set <math>N = \{x: f(x) \neq \lim_{n \to \infty} f(x) \}</math>. | |||
A sequence of measurable functions <math>\{f_n \}</math> converges to <math> f<\math>pointwise almost everywhere if <math> f_m (x) \to f(x)<\math> for almost every <math> x </math>, or <math> \mu(N) =0< /math>. | |||
Revision as of 06:55, 18 December 2020
Definition
Denote our measure space as .
Define the set .
A sequence of measurable functions converges to Failed to parse (unknown function "\math"): {\displaystyle f<\math>pointwise almost everywhere if <math> f_m (x) \to f(x)<\math> for almost every <math> x }
, or <math> \mu(N) =0< /math>.
