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Analogs of the Lebesgue measure in spaces of sequences and classes of functions integrable with respect to these measures. (English. Russian original) Zbl 1460.28011

J. Math. Sci., New York 252, No. 1, 36-42 (2021); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 151, 37-44 (2018).
For any \(p\in[1,\infty]\) let \(\ell^p\) be the space of real \(p\)-summable sequences endowed with its topology of coordinate-wise convergence. The main results of the paper under review provide quite specific constructions of non-zero translation-invariant \(\sigma\)-additive regular Borel measures \(\lambda_p\) on \(\ell^p\). These measures are not \(\sigma\)-finite, since \(\ell^p\) is an infinite-dimensional vector space, and one also proves that the measure \(\lambda_p\) is not complete. Additional facts are established: The space of bounded functions that are continuous on \(\ell^p\) and vanish outside a set of finite measure give rise to a dense subset of the Banach space \(L^q(\lambda_p)\), and this Banach space is not separable. If \(1\le p<\infty\), then the Borel \(\sigma\)-algebra of \(\ell^p\) is the same as the Borel \(\sigma\)-algebra associated to the usual Banach space topology of this space, and the same as the Borel \(\sigma\)-algebra associated to the topology of uniform convergence. For \(p=\infty\), there exists a continuous function on \(\ell^p\) which is not Borel measurable. One also carefully compares these results with the earlier literature.

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
47D08 Schrödinger and Feynman-Kac semigroups
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References:

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