Convergence of measurable random functions

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Using the theorem of Fréchet and Kolmogorov about compact subsets of the space Lp[0, 1] and Prohorov's theorem about the convergence of measures defined on a complete metric space we proved in [2] the following theorem: Let ξ1(t),…, ξn,(/)>… and ξ (t) be measurable random processes (0 ≤ t ≤ 1) and suppose that there exist numbers C and p ≥ 1 such that E|ξn(t)p| ≤ C for all n and t. If E|ξn(t)p|→ £|£(0lp for all /and if for any finite set {t1,…, tk] c [0, 1] the joint distribution of £„(f,),…, £„(tk) converges to the joint distribution of ξ (tn),…, ξ (tk), then the distribution of F(ξn) converges to the distribution of F(ξ) for any continuous functional F on LP,[0, 1]. In this paper this theorem is generalized to random measurable functions. The results of the present paper are related to the results of [1], [3], [4].

Original languageEnglish
Pages (from-to)322-325
Number of pages4
JournalProceedings of the American Mathematical Society
Issue number2
StatePublished - May 1979
Externally publishedYes


  • Convergence of joint distributions
  • Convergence of measures
  • Measurable random functions


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