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Uniquely Determined Uniform Probability on the Natural Numbers

Uniquely Determined Uniform Probability on the Natural Numbers In this paper, we address the problem of constructing a uniform probability measure on $${\mathbb {N}}$$ N . Of course, this is not possible within the bounds of the Kolmogorov axioms, and we have to violate at least one axiom. We define a probability measure as a finitely additive measure assigning probability 1 to the whole space, on a domain which is closed under complements and finite disjoint unions. We introduce and motivate a notion of uniformity which we call weak thinnability, which is strictly stronger than extension of natural density. We construct a weakly thinnable probability measure, and we show that on its domain, which contains sets without natural density, probability is uniquely determined by weak thinnability. In this sense, we can assign uniform probabilities in a canonical way. We generalize this result to uniform probability measures on other metric spaces, including $${\mathbb {R}}^n$$ R n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

Uniquely Determined Uniform Probability on the Natural Numbers

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References (14)

Publisher
Springer Journals
Copyright
Copyright © 2015 by The Author(s)
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
DOI
10.1007/s10959-015-0611-2
Publisher site
See Article on Publisher Site

Abstract

In this paper, we address the problem of constructing a uniform probability measure on $${\mathbb {N}}$$ N . Of course, this is not possible within the bounds of the Kolmogorov axioms, and we have to violate at least one axiom. We define a probability measure as a finitely additive measure assigning probability 1 to the whole space, on a domain which is closed under complements and finite disjoint unions. We introduce and motivate a notion of uniformity which we call weak thinnability, which is strictly stronger than extension of natural density. We construct a weakly thinnable probability measure, and we show that on its domain, which contains sets without natural density, probability is uniquely determined by weak thinnability. In this sense, we can assign uniform probabilities in a canonical way. We generalize this result to uniform probability measures on other metric spaces, including $${\mathbb {R}}^n$$ R n .

Journal

Journal of Theoretical Probability – Springer Journals

Published: Apr 12, 2015

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