Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with Respect to These Measure
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ANALOGS OF THE LEBESGUE MEASURE IN SPACES OF SEQUENCES AND CLASSES OF FUNCTIONS INTEGRABLE WITH RESPECT TO THESE MEASURES D. V. Zavadskii
UDC 517.982, 517.983
Abstract. We examine translation-invariant measures on Banach spaces lp , where p ∈ [1, ∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp , where p ∈ [1, ∞]. For p ∈ [1, ∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties. Keywords and phrases: translation-invariant measure, topology of pointwise convergence, Borel σalgebra, space of integrable functions, approximation of integrable functions by continuous functions. AMS Subject Classification: 28C20, 81Q05, 47D08
1. Introduction. Translation-invariant (or shift-invariant) measures are used for the study of solutions to differential equations based on mathematical expectations of functionals of random walks. Application of translation-invariant measures to representations of solutions to differential equations can be found in [2, 4], where strongly continuous one-parameter operator semigroups that solve Cauchy problems for the diffusion equation, the fractional diffusion equation, and the Schr¨odinger equation with various Hamiltonians were obtained by averaging of random one-parameter families of operators of shifts by vectors of the coordinate space by measures defined on a set of shift operators. Thus approach to the study of properties of solutions to differential equations for functions on infinitedimensional spaces requires the analysis of measures on infinite-dimensional spaces that are invariant under shifts by vectors of this space or under other transformation groups (see [6]). Translation-invariant measures are also useful in the study of Feynman path integrals (see [3, 7]). It was proved in [9] that there is no Lebesgue measure on any infinite-dimensional topological vector space, i.e., there is no nonzero, σ-additive, σ-finite, shift-invariant measure on the σ-algebra of Borel subsets of an infinite-dimensional topological vector space. In this relation, problems on the existence of measures on infinite-dimensional topological vector space that satisfy only several properties of the Lebesgue measure (see [1, 5, 6, 8]). In this paper, we examine measures on Banach spaces l
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