Continuous Selections for Vector Measures (original) (raw)

Continuous Selections for Vector Measures Author ( s )

2007

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. A vector measure is a many to one map; it maps many measurable sets onto the same point. A selection for a vector measure is a function which assigns to each point in the range of the vector measure only one measurable set which is mapped onto the point. The existence of a co...

Selections and their absolutely continuous invariant measures

Journal of Mathematical Analysis and Applications, 2014

Let I = [0, 1] and consider disjoint closed regions G 1 , ...., Gn in I × I and subintervals I 1 , ......, In, such that G i projects onto I i. We define the lower and upper maps τ 1 , τ 2 by the lower and upper boundaries of G i , i = 1, ...., n, respectively. We assume τ 1 , τ 2 to be piecewise monotonic and preserving continuous invariant measures µ 1 and µ 2 , respectively. Let F (1) and F (2) be the distribution functions of µ 1 and µ 2. The main results shows that for any convex combination F of F (1) and F (2) we can find a map η with values between the graphs of τ 1 and τ 2 (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of multi-valued maps to random maps.

On maximal ranges of vector measures for subsets and purification of transition probabilities

Proceedings of the American Mathematical Society, 2011

Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the σ \sigma -field into a Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two 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.

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.

The Vector Measures Whose Range Is Strictly Convex

Journal of Mathematical Analysis and Applications, 1999

In this paper we resume the most important results that we obtained in our papers [1,2,5,6,7] concerning a broad class of measures that we defined in dealing with a bangbang control problem. Let M be the σ−algebra of the Lebesgue measurable subsets of [0, 1] and µ : M → R n be a non-atomic vector measure. A well known Theorem of Lyapunov (see [11]) states that the range of µ, defined by R(µ) = {µ(E) : E ∈ M}, is closed and convex or, equivalently, that given a measurable function ρ with values in [0, 1] there exists a set E in M such that (*) X ρ dµ = µ(E). Lyapunov's Theorem has been widely applied in bang-bang control theory [10] and, more recently, in some non-convex problems of the Calculus of Variations [3]. As an example we mention the following bang-bang existence result:

On semi-discrete sub-partitions of vector-valued measures

arXiv: Optimization and Control, 2020

We introduce a concept of optimal transport for vector-valued measures and its dual formulation. In this note we concentrate on the semi-discrete case and show some fundamental differences between the scalar and vector cases. A manifestation of this difference is the possibility of non-existence of optimal solution for the dual problem for feasible primer problems.

A Generalization of Lyapounov's Convexity Theorem to Measures with Atoms

Proceedings of The American Mathematical Society, 1987

The distance from the convex hull of the range of an n-dimensional vector-valued measure to the range of that measure is no more than an/2, where a is the largest (one-dimensional) mass of the atoms of the measure. The case a = 0 yields Lyapounov's Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.