modes of convergence of sequences of measurable functions

Let (X,𝔅,μ) be a measure spaceMathworldPlanetmath, fn:X[-,] be measurable functionsMathworldPlanetmath for every positive integer n, and f:X[-,] be a measurable function. The following are modes of convergence of {fn}:

  • {fn} converges almost everywhere to f if μ(X-{xX:limnfn(x)=f(x)})=0

  • {fn} converges almost uniformly to f if, for every ε>0, there exists Eε𝔅 with μ(X-Eε)<ε and {fn} converges uniformly to f on Eε

  • {fn} converges in measure to f if, for every ε>0, there exists a positive integer N such that, for every positive integer nN, μ({xX:|fn(x)-f(x)|ε})<ε.

  • If, in , f and each fn are also Lebesgue integrableMathworldPlanetmath, {fn} converges in L1(μ) to f if limnX|fn-f|𝑑μ=0.

A lot of theorems in real analysis ( deal with these modes of convergence. For example, Fatou’s lemma, Lebesgue’s monotone convergence theoremMathworldPlanetmath, and Lebesgue’s dominated convergence theorem give conditions on sequencesMathworldPlanetmath of measurable functions that converge almost everywhere under which they also converge in L1(μ). Also, Egorov’s theorem that, if μ(X)<, then convergence almost everywhere implies almost uniform convergenceMathworldPlanetmath.

Title modes of convergence of sequences of measurable functions
Canonical name ModesOfConvergenceOfSequencesOfMeasurableFunctions
Date of creation 2013-03-22 16:14:05
Last modified on 2013-03-22 16:14:05
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Definition
Classification msc 28A20
Related topic TravelingHumpSequence
Related topic VitaliConvergenceTheorem
Defines converges almost everywhere
Defines convergence almost everywhere
Defines converges almost uniformly
Defines almost uniform convergence
Defines converges in measure
Defines convergence in measure
Defines converges in L1(μ)
Defines L1(μ) convergence