## The Annals of Probability

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

### Partitioning General Probability Measures

#### 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