Three problems of aronszajn in measure theory

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.

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.

Publicacions Matemàtiques, 1991

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.

American Mathematical Monthly, 1972

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where &quot;almost everywhere&quot; refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure $\mu$. In particular we show that the differentiability properties of Lipschitz functions at $\mu$-almost every point are related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures. In the process we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.

This preliminary chapter deals with basic concepts and results from both measure theory and functional analysis as much of the theory put forward in this thesis relies on standard results from these two highly interconnected fields of mathematics. 3 Note that, if inf s v(s) > 0 then tightness of the family {µ n : n ∈ N} is a consequence of µ n Cv =⇒ µ. 2. MEASURE-VALUED DIFFERENTIATION This chapter is devoted to a detailed analysis of the concept of measure-valued differentiation and its applicability. New results will be established, by combining functional analytic and measure theoretical techniques and some applications will be provided.

Transactions of the American Mathematical Society, 1982

We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let {n¡)¡ei be a projective system of Radon measures, we use the Loeb-measure L(jiE) for an infinite E £ */ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov's Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hubert spaces (Sazonov*s Theorem), and extend cylindrical measures on Hubert spaces to o-additive measures on Banach spaces (Gross' Theorem).