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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 459-464 (mai 2012)
Doi : 10.1016/j.crma.2012.03.010
Received : 26 January 2012 ;  accepted : 7 Mars 2012
On the regular convergence of multiple integrals of locally Lebesgue integrable functions over  
Sur la convergence régulière dʼintégrales multiples définies sur   localement intégrables au sens de Lebesgue
 

Ferenc Móricz
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary 

Abstract

Let the function   be such that  , where   is a fixed integer. We investigate the convergence behavior of the m -multiple integral
∫0v1∫0v2…∫0vmf(t1,t2,…,tm)dt1dt2…dtmas min{v1,v2,…,vm}→∞, while using two notions of convergence: the one in Pringsheimʼs sense and the one in the regular sense. For the sake of brevity, we present our main result in the case   as follows: If   and the double integral (⁎) converges regularly, then the finite limits   and   exist uniformly in  , respectively, and  . This can be considered as a generalized version of Fubiniʼs theorem on successive integration in the case when  .

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Résumé

Soit   telle que  , où m est un entier fixé. On étudie la convergence de lʼintégrale multiple dʼordre m ,   quand  , en utilisant deux méthodes de convergence, lʼune au sens de Pringsheim, et lʼautre au sens régulier. Pour simplifier on présente notre résultat fondamental pour  , de la façon suivante : Si   et si lʼintégrale double converge régulièrement, alors les limites finies   et   existent uniformément dans  , respectivement, et on a  . Ceci peut être considéré comme une généralisation du théorème de Fubini concernant lʼintégration successive au cas où  .

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