scholarly journals When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

1982 ◽  
Vol 33 (3) ◽  
pp. 374-385 ◽  
Author(s):  
Thomas E. Armstrong
Keyword(s):  
Borel Measure ◽  
Convex Set ◽  
Additive Measure ◽  
Dimensional Simplex ◽  
Finitely Additive Measure ◽  
Regular Algebra ◽  
Finite Dimensional ◽  
Regular Sets ◽  
Dirac Measures

AbstractIt is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

scholarly journals Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

A Remark on Disintegrations With Almost All Components Non -Additive

1979 ◽  
Vol 31 (4) ◽  
pp. 786-788 ◽  
Author(s):  
Nghiem Dang-Ngoc
Keyword(s):  
Real Function ◽  
Measure Space ◽  
Gibbs States ◽  
List Type ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Coin Tossing ◽  
Definition Of ◽  
Almost All

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:(i) ∀x ∊ X , σx(.) is a finitely additive measure .(ii) ∀A ∊ , σ. (A) is constant on each -atom.(iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and (iv)σx(B) = 1 if x ∊ B ∊ .

Dynamical Properties of Quadratic Homeomorphisms of a Finite-Dimensional Simplex

2020 ◽  
Vol 245 (3) ◽  
pp. 398-402
Author(s):  
R. N. Ganikhodzhaev ◽  
M. A. Tadzhieva ◽  
D. B. Eshmamatova
Keyword(s):  
Dimensional Simplex ◽  
Finite Dimensional ◽  
Dynamical Properties

scholarly journals Some remarks concerning finitely Subadditive outer measures with applications

1998 ◽  
Vol 21 (4) ◽  
pp. 653-669 ◽  
Author(s):  
John E. Knight
Keyword(s):  
General Theory ◽  
Outer Measure ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Set Functions ◽  
Relationship Of ◽  
The Relationship ◽  
General Method

The present paper is intended as a first step toward the establishment of a general theory of finitely subadditive outer measures. First, a general method for constructing a finitely subadditive outer measure and an associated finitely additive measure on any space is presented. This is followed by a discussion of the theory of inner measures, their construction, and the relationship of their properties to those of an associated finitely subadditive outer measure. In particular, the interconnections between the measurable sets determined by both the outer measure and its associated inner measure are examined. Finally, several applications of the general theory are given, with special attention being paid to various lattice related set functions.

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

2015 ◽  
Vol 99 (2) ◽  
pp. 145-165 ◽  
Author(s):  
G. BEER ◽  
J. VANDERWERFF
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Separation Theorems ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Weakly Closed ◽  
Separation Of Convex Sets ◽  
Real Linear ◽  
Closed Convex Set ◽  
Special Case

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

scholarly journals The nonlinear limit control of EDSQOs on finite dimensional simplex

Automatika ◽  
2019 ◽  
Vol 60 (4) ◽  
pp. 404-412 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Shahrum Shah Abdullah ◽  
Sherzod Turaev ◽  
Raini Hassan
Keyword(s):  
Dimensional Simplex ◽  
Finite Dimensional

scholarly journals Measures on coallocation and normal lattices

1992 ◽  
Vol 15 (4) ◽  
pp. 701-718
Author(s):  
Jack-Kang Chan
Keyword(s):  
Additive Measure ◽  
Regular Measure ◽  
Finitely Additive Measure

Letℒ1andℒ2be lattices of subsets of a nonempty setX. Supposeℒ2coallocatesℒ1andℒ1is a subset ofℒ2. We show that anyℒ1-regular finitely additive measure on the algebra generated byℒ1can be uniquely extended to anℒ2-regular measure on the algebra generated byℒ2. The case whenℒ1is not necessary contained inℒ2, as well as the measure enlargement problem are considered. Furthermore, some discussions on normal lattices and separation of lattices are also given.

scholarly journals Smoothness conditions on measures using Wallman spaces

1999 ◽  
Vol 22 (4) ◽  
pp. 713-726
Author(s):  
Charles Traina
Keyword(s):  
General Problem ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Finitely Additive Measures ◽  
Necessary And Sufficient ◽  
Additive Measures

In this paper,Xdenotes an arbitrary nonempty set,ℒa lattice of subsets ofXwith∅,X∈ℒ,A(ℒ)is the algebra generated byℒandM(ℒ)is the set of nontrivial, finite, and finitely additive measures onA(ℒ), andMR(ℒ)is the set of elements ofM(ℒ)which areℒ-regular. It is well known that anyμ∈M(ℒ)induces a finitely additive measureμ¯on an associated Wallman space. Wheneverμ∈MR(ℒ),μ¯is countably additive.We consider the general problem of givenμ∈MR(ℒ), how do properties ofμ¯imply smoothness properties ofμ? For instance, what conditions onμ¯are necessary and sufficient forμto beσ-smooth onℒ, or stronglyσ-smooth onℒ, or countably additive? We consider in discussing these questions either of two associated Wallman spaces.

Sign in / Sign up

Export Citation Format

Share Document