The Annals of Probability

Partitioning General Probability Measures

Theodore P. Hill

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Abstract

Suppose $\mu_1,\ldots,\mu_n$ are probability measures on the same measurable space $(\Omega, \mathscr{F})$. Then if all atoms of each $\mu_i$ have mass $\alpha$ or less, there is a measurable partition $A_1,\ldots, A_n$ of $\Omega$ so that $\mu_i(A_i) \geq V_n(\alpha)$ for all $i = 1,\ldots, n$, where $V_n(\cdot)$ is an explicitly given piecewise linear nonincreasing continuous function on [0, 1]. Moreover, the bound $V_n(\alpha)$ is attained for all $n$ and all $\alpha$. Applications are given to $L_1$ spaces, to statistical decision theory, and to the classical nonatomic case.

Article information

Source
Ann. Probab., Volume 15, Number 2 (1987), 804-813.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992173

Digital Object Identifier
doi:10.1214/aop/1176992173

Mathematical Reviews number (MathSciNet)
MR885145

Zentralblatt MATH identifier
0625.60004

JSTOR
links.jstor.org

Subjects
Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}
Secondary: 28A99: None of the above, but in this section 60E15: Inequalities; stochastic orderings 62C20: Minimax procedures

Keywords
Optimal-partitioning inequalities atomic probability measures cake-cutting fair division problems minimax decision rules

Citation

Hill, Theodore P. Partitioning General Probability Measures. Ann. Probab. 15 (1987), no. 2, 804--813. doi:10.1214/aop/1176992173. https://projecteuclid.org/euclid.aop/1176992173


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