Pantsulaia, Gogi Duality of measure and category in infinite-dimensional separable Hilbert space \(\ell _{2}\). (English) Zbl 1011.28010 Int. J. Math. Math. Sci. 30, No. 6, 353-363 (2002). The author constructs a concrete nontrivial \(\sigma\)-finite Borel measure \(\lambda\) on the infinite-dimensional separable Hilbert space \(\ell_2\) to show that there is a Borel subset \(Y\) of \(\ell_2\) with \(\lambda(Y)>0\) such that for any \(\delta>0\) there is a \(y\in\ell_2\) satisfying \(\|y\|<\delta\) and \(Y\cap (Y+y)=\emptyset\). This result states that an analogy of the Oxtoby duality principle is not valid for the measure \(\lambda\) and the Baire category in the Hilbert space \(\ell_2\). Reviewer: Jun Kawabe (Wakasato / Nagano) Cited in 1 Document MSC: 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 54E52 Baire category, Baire spaces 28A35 Measures and integrals in product spaces Keywords:duality of measure and category; Borel invariant measure; Oxtoby duality principle PDFBibTeX XMLCite \textit{G. Pantsulaia}, Int. J. Math. Math. Sci. 30, No. 6, 353--363 (2002; Zbl 1011.28010) Full Text: DOI EuDML Link