A note on the weak convergence of continuously integrable sequences

2021/08/07 by Gane Samb Lô, Lo, Gane Samb, Aladji Babacar Niang +1
Economics, Econometrics and Finance · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2108.03484

openalex publication_date 2021/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the σ(L¹,L^∞) topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the ideas beneath the proof of prohorov's theorem where the continuous integrability replaces the uniform or asymptotic tightness.

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