A note on measure extension problem of ℓ-group-valued measures

Algebraic ramifications of the common extension problem for group-valued measures

Let G be an Abelian group and let μ : A → G and ν : B → G be finitely additive measures (charges) defined on fields A and B of subsets of a set X. It is assumed that μ and ν agree on A∩B, i.e. they are consistent. The existence of common extensions of μ and ν is investigated, and conditions on A and B facilitating such extensions are given. 0. Introduction. We consider the following problem: Le...

Decomposition of group-valued measures on orthoalgebras

We present a general decomposition theorem for a positive inner regular finitely additive measure on an orthoalgebra L with values in an ordered topological group G, not necessarily commutative. In the case where L is a Boolean algebra, we establish the uniqueness of such a decomposition. With mild extra hypotheses on G, we extend this Boolean decomposition, preserving the uniqueness, to the ca...

Hyperreal-Valued Probability Measures Approximating a Real-Valued Measure

A Note on Invariant Measure on the Local Gauge Group

In this paper we investigate the problem of the existence of invariant measures on the local gauge group. We prove that it is impossible to define a finite translationally invariant measure on the local gauge group C∞(Rn, G)(where G is an arbitrary matrix Lie group). Functional integral over the local gauge group is a commonly employed technique in the study of gauge field theories. Its earlies...

Random Measures and Measure-Valued Processes

The question of “levels of selection” has received a great deal of attention and controversy in the biological literature but relatively little attention in the mathematical literature. In this lecture we brie y review some background and formulate some models of multitype populations undergoing mutation and selection including multilevel selection. The main tool is a related dual processes whi...