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单词 ModesOfConvergenceOfSequencesOfMeasurableFunctions
释义

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 (http://planetmath.org/BibliographyForRealAnalysis) 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.

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更新时间:2025/5/4 8:06:43