Measures in banach spaces are determined by their values on balls (original) (raw)

Existence and uniqueness of translation invariant measures in separable Banach spaces

Functiones et Approximatio Commentarii Mathematici, 2014

It is shown that for the vector space 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 σ-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 σ-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 σ-finite Borel measure is solved positively. The uniqueness problem for non-σ-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 µ 0 B on X, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of µ 0 B and whose values on non-degenerate rectangles coincide with their volumes.

On Ordinary and Standard "Lebesgue Measures" in Separable Banach Spaces

2013

By using results from a paper [G.R. Pantsulaia, On ordinary and standard Lebesgue measures on R ∞ , Bull. Pol. Acad. Sci. Math. 57 (3-4) (2009), 209-222] and an approach based in a paper [T. Gill, A.Kirtadze, G.Pantsulaia , A.Plichko, The existence and uniqueness of translation invariant measures in separable Banach spaces, Functiones et Approximatio, Commentarii Mathematici, 16 pages, to appear ], a new class of translation-invariant quasi-finite Borel measures (the so called, ordinary and standard "Lebesgue Measures") in an infinite-dimensional separable Banach space X is constructed and some their properties are studied in the present paper. Also, various interesting examples of generators of two-sided (left or right) shy sets with domain in non-locally compact Polish Groups are considered.

Remarks on the semivariation of vector measures with respect to Banach spaces

Bulletin of the Australian Mathematical Society, 2007

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

A note on ball-covering property of Banach spaces

Journal of Mathematical Analysis and Applications, 2010

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere S X of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X * of X is w * separable, then for every ε > 0 there exist a 1 + ε equivalent norm on X, and an R > 0 such that in this new norm S X admits a ball-covering by countably many balls of radius R. Namely, we show that R = R(ε) can be taken arbitrarily close to (1 + ε)/ε, and that for X = 1 [0, 1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε → 0.

On some open problems involving range of vector measures

Filomat, 2012

The close connection between the geometry of a Banach space and the properties of vector measures acting into it is now fairly well-understood. The present paper is devoted to a discussion of some of these developments and certain problems arising naturally in this circle of ideas which are either open or have been partially resolved. Emphasis shall be laid mainly on those aspects of this theory which involve properties of the range of these vector measures.

On Some Aspects of Vector Measures

Advances in Mathematics: Scientific Journal

This paper addresses some properties of vector measures (Banach space valued measures) as well as topological results on some spaces of vector measures of bounded variation.

VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE

Function Spaces - Proceedings of the Sixth Conference, 2003

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Köthe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions f X : Ω → [0, ∞) are in the lattice E. The notion of E-variation of a measure -which allows to recover the pvariation (for E = L p ), Φ-variation (for E = L Φ ) and the general notion introduced by Gresky and Uhl-is introduced. The space of measures of bounded E-variation V E (X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X)) * = V E (X * ), that V E (X) can be identified with space of cone absolutely summing operators from E into X and that E(X) = V E (X) if and only if X has the RNP property.

The effect of projections on fractal sets and measures in Banach spaces

Ergodic Theory and Dynamical Systems, 2006

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finitedimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the 'thickness exponent' of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263-1275). More precisely, let X be a compact subset of a Banach space B with thickness exponent τ and Hausdorff dimension d. Let M be any subspace of the (locally) Lipschitz functions from B to R m that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f ∈ M, the Hausdorff dimension of f (X) is at least min{m, d/(1 + τ)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + τ) can be improved to 1/(1 + τ/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when τ = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case τ > 0.