Uniquely Determined Uniform Probability on the Natural Numbers
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- 作者：Timber Kerkvliet ; Ronald Meester
- 关键词：Uniform probability ; Foundations of probability ; Kolmogorov axioms ; Finite additivity
- 刊名：Journal of Theoretical Probability
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：29
- 期：3
- 页码：797-825
- 全文大小：549 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Probability Theory and Stochastic Processes
Statistics - 出版者：Springer Netherlands
- ISSN：1572-9230
- 卷排序：29
文摘
In this paper, we address the problem of constructing a uniform probability measure on \({\mathbb {N}}\). 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 \({\mathbb {R}}^n\).