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Duality of measure and category in infinite-dimensional separable Hilbert space \(\ell _{2}\). (English) Zbl 1011.28010

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 \(\ell_2\).

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
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