Rado's Theorem for the Loeb Space of an Internal ∗-Finitely Additive Measure Space

1991 ◽  
Vol 112 (1) ◽  
pp. 203
Author(s):  
Bosko Zivaljevic
Keyword(s):  
Measure Space ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Loeb Space

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 ∊ .

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 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.

On Egoroff's theorems on finite monotone non-additive measure space

2005 ◽  
Vol 153 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Jun Li ◽  
Masami Yasuda
Keyword(s):  
Measure Space ◽  
Additive Measure

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.

scholarly journals On maximal measures with respect to a lattice

1991 ◽  
Vol 14 (1) ◽  
pp. 93-98
Author(s):  
James Camacho
Keyword(s):  
Additive Measure ◽  
Finitely Additive Measure ◽  
Normal Lattice ◽  
Smooth Measures

Outer measures are used to obtain measures that are maximal with respect to a normal lattice. Alternate proofs are then given extending the measure theoretic characterizations of a normal lattice to an arbitrary, non-negative finitely additive measure on the algebra generated by the lattice. Finally these general results are used to considerσ-smooth measures with respect to the lattice when further conditions on the lattice hold.

Measures Below Outer Measures

1994 ◽  
Vol 37 (2) ◽  
pp. 270-271
Author(s):  
Stephen Watson
Keyword(s):  
Simple Proof ◽  
Cantor Set ◽  
Outer Measure ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Finite Set

AbstractWe give a simple proof that, for any ॉ > 0, there is an outer measure μ* on a finite set X such that, for any measure Thus there is a non-zero outer (finitely subadditive) measure v* on the clopen subsets of the Cantor set such that, if v ≤ v* is a finitely additive measure on the clopen subsets of the Cantor set, then v ≡ 0.

scholarly journals AN EIGENSPACE OF LARGE DIMENSION FOR A HECKE ALGEBRA ON AN BUILDING

2011 ◽  
Vol 91 (1) ◽  
pp. 29-54 ◽  
Author(s):  
A. M. MANTERO ◽  
A. ZAPPA
Keyword(s):  
Hecke Algebra ◽  
Large Dimension ◽  
Difference Operators ◽  
Additive Measure ◽  
Laplace Operators ◽  
Finitely Additive Measure ◽  
Poisson Transform ◽  
Affine Building ◽  
Joint Eigenfunctions ◽  
Constant Coefficients

AbstractLet Δ be an affine building of type $\tilde A_2$ and let 𝔸 be its fundamental apartment. We consider the set 𝕌0 of vertices of type 0 of 𝔸 and prove that the Hecke algebra of all W0-invariant difference operators with constant coefficients acting on 𝕌0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than ∣W0 ∣. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely additive measure on the maximal boundary Ω of Δ.

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