soxumis saxelmwifo universitetis S r o m e b i VII

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaGOGI PANTSULAIAON A RIEMANN INTEGRABILITY OF FUNCTIONS DEFINEDON INFINITE-DIEMENSIONAL RECTANGLESAbstract. We announce a result asserted that for a Riemann integrablefunction f defined on the infinite-dimensional rectanglei1[ a , b]ОВ , an infinite-dimensional version of the Weyl theorem isiivalid. This fact allows us to give an effective algorithm for a calculationof the Riemann integralтҐХ[ ai, bi]i=1( R) f ( x) dl( x), where l denotes an infinite-dimensional‗Lebesgue measure‘ constructed by R.Baker in 1991.2010 Mathematical Subject Classification: Primary 28Axx,28Cxx, 28Dxx; Secondary 28C20, 28D10, 28D99.Key words and phrases: Riemann integrability, Lebesguemeasure, infinite-dimensional rectangles1. IntroductionVarious questions combinatorial or discrete type frequently arise indifferent domains of modern mathematics (especially, in Mathematicalanalysis, Measure theory, Differential equations, Game theory, Set Theory,Graph Theory etc.) and are important from the theoretical viewpointand from the view-point of their numerous applications. In particular,these questions play a key role in applications of algorithms andcomputer science. For example, the notion of a equidistributed, or uniformlydistributed sequences ( an)nО Nin an interval [ ab , ] describes acertain discrete mathematical structure which has various interesting44