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

D. V. Zavadskii

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Quantum probability, Tome 151 (2018), pp. 37-44

Résumé

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.

Détail

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