scholarly journals On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity

2008 ◽  
Vol 2008 ◽  
pp. 1-13
Author(s):  
Charles Traina
Keyword(s):  
Outer Measure ◽  
Set Functions ◽  
Finite Set ◽  
Covering Class ◽  
Set Function

Given a nonempty abstract set , and a covering class , and a finite, finitely subadditive outer measure , we construct an outer measure and investigate conditions for to be submodular. We then consider several other set functions associated with and obtain conditions for equality of these functions on the lattice generated by . Lastly, we describe a construction of a finite, finitely subadditive outer measure given an arbitrary family of subsets, , of and a nonnegative, finite set function defined on .

Covering Classes of Residues

1967 ◽  
Vol 19 ◽  
pp. 514-519 ◽  
Author(s):  
James H. Jordan
Keyword(s):  
Finite Set ◽  
Covering Class ◽  
Ordered Pairs

A set of ordered pairs of integers {(ai, mi)} is said to cover the integers if each integer x satisfies the congruence x ≡ ai (mod mi) for some i. We may assume that the mi are positive. Trivially {(0, 1)} covers, as does {(0, m), (1, m), (2, m), … , (m — 1, m)}. In order to arrive at some non-trivial problems concerning covers, the following definition is given: A finite set of ordered pairs of integers with mi > 1 and mi ≠ mj if i ≠ j, is called a covering class of residues if every integer x satisfies the congruence x ≡ ai (mod mi) for some i.

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.

scholarly journals Wilansky's query on outer measures

1985 ◽  
Vol 31 (3) ◽  
pp. 325-328
Author(s):  
Choo-Whan Kim
Keyword(s):  
Outer Measure ◽  
Finite Set ◽  
Infinite Set

On a set X, let μ* be an outer measure and μ the measure induced by μ*. We show that if X is a finite set, then the measure μ is saturated. We give two examples of non-regular outer measures on an infinite set X which induce non-saturated and saturated measures, respectively. These answer a query posed by Wilansky.

A proof-theoretic characterization of the primitive recursive set functions

1992 ◽  
Vol 57 (3) ◽  
pp. 954-969 ◽  
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Elementary Proof ◽  
Weak Version ◽  
Definable Set ◽  
Set Functions ◽  
Theoretic Characterization ◽  
Universe Of Sets ◽  
Set Function ◽  
Primitive Recursive

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.

On uniform boundedness properties in exhausting additive set function spaces

1981 ◽  
Vol 90 (1-2) ◽  
pp. 175-184 ◽  
Author(s):  
Aníbal Moltó
Keyword(s):  
Topological Group ◽  
Duality Theory ◽  
Uniform Boundedness ◽  
Baire Category ◽  
Boolean Algebras ◽  
Topological Spaces ◽  
Baire Category Theorem ◽  
Set Functions ◽  
Category Theorem ◽  
Set Function

SynopsisValdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.

scholarly journals Interaction transform for bi-set functions over a finite set

2006 ◽  
Vol 176 (16) ◽  
pp. 2279-2303 ◽  
Author(s):  
F LANGE ◽  
M GRABISCH
Keyword(s):  
Set Functions ◽  
Finite Set

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 Integration with respect to the product of a measure and a closable set function

1989 ◽  
Vol 115 ◽  
pp. 47-62
Author(s):  
Brian Jefferies
Keyword(s):  
Operator Equations ◽  
Schrödinger’S Equation ◽  
Set Functions ◽  
Scalar Measure ◽  
Set Function ◽  
Schrödinger's Equation

The notion of a set function being closable with respect to a family of measures was introduced in [1] and applied there to the integration of Feynman-Kac functionals. In [2], it is shown how the interchange of integrals for the product of an operator valued measure and a scalar measure can be used to solve operator equations associated with the perturbation of semigroups. The purpose of this article is to develop the analogous machinery for closable set functions with the view of applying it to Schrödinger’s equation.

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