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Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2018, Volume 151, Pages 37–44 (Mi into338)  

Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with respect to These Measures

D. V. Zavadskii

Moscow Institute of Physics and Technology (State University)

Abstract: We examine translation-invariant measures on Banach spaces $l_p$, where $p\in[1,\infty]$. We construct analogs of the Lebesgue measure on Borel $\sigma$-algebras generated by the topology of pointwise convergence ($\sigma$-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not $\sigma$-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 $\sigma$-algebras corresponding to various topologies in the spaces $l_p$, where $p\in[1,\infty]$. For $p\in [1, \infty)$, we prove the coincidence of Borel $\sigma$-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel $\sigma$-algebra corresponding to the topology of pointwise convergence. We also verify that the space $l_\infty$ does not possess similar properties.

Keywords: translation-invariant measure, topology of pointwise convergence, Borel $\sigma$-algebra, space of integrable functions, approximation of integrable functions by continuous functions

Full text: PDF file (185 kB)

Bibliographic databases:

Document Type: Article
UDC: 517.982, 517.983
MSC: 28C20, 81Q05, 47D08

Citation: D. V. Zavadskii, “Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with respect to These Measures”, Quantum probability, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 151, VINITI, Moscow, 2018, 37–44

Citation in format AMSBIB
\Bibitem{Zav18}
\by D.~V.~Zavadskii
\paper Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with respect to These Measures
\inbook Quantum probability
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.
\yr 2018
\vol 151
\pages 37--44
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into338}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3903364}


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