Functiones et Approximatio Commentarii Mathematici

Existence and uniqueness of translation invariant measures in separable Banach spaces

Tepper Gill, Aleks Kirtadze, Gogi Pantsulaia, and Anatolij Plichko

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Abstract

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 2 (2014), 401-419.

Dates
First available in Project Euclid: 26 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1403811851

Digital Object Identifier
doi:10.7169/facm/2014.50.2.12

Mathematical Reviews number (MathSciNet)
MR3229068

Zentralblatt MATH identifier
1296.28015

Subjects
Primary: 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]

Keywords
invariant Borel measure admissible translation Banach space Polish linear space Markushevich basis

Citation

Gill, Tepper; Kirtadze, Aleks; Pantsulaia, Gogi; Plichko, Anatolij. Existence and uniqueness of translation invariant measures in separable Banach spaces. Funct. Approx. Comment. Math. 50 (2014), no. 2, 401--419. doi:10.7169/facm/2014.50.2.12. https://projecteuclid.org/euclid.facm/1403811851


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