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A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures
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  • Published: May 1987

A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures

  • Theodore P. Hill1 

Probability Theory and Related Fields volume 75, pages 143–147 (1987)Cite this article

  • 47 Accesses

  • 5 Citations

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Summary

If μ 1, ... , μ η are non-atomic probability measures on the same measurable space (S, ℱ), then there is an ℱ-measurable partition {A i } n i = 1 of S so that μ i (A i )≧(n − 1 + m)−1 for all i=1, ..., n, where \(m = \left\| {\mathop \Lambda \limits_{i = 1}^n \mu _i } \right\|\) is the total mass of the largest measure dominated by each of the μ i ’S; moreover, this bound is attained for all n≧1 and all m in [0, 1]. This result is an analog of the bound (n+1-M) -1of Elton et al. [5] based on the mass M of the supremum of the measures; each gives a quantative generalization of a well-known cake-cutting inequality of Urbanik [10] and of Dubins and Spanier [2].

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References

  1. Demko, S., Hill, T.: Equitable distribution of indivisible objects. Preprint (1985)

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  4. Elton, J., Hill, T.: A generalization of Lyapounov's convexity theorem to measures with atoms. Proc. Am. Math. Soc., 99, 297–304 (1987)

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  5. Elton, J., Hill, T., Kertz, R.: Optimal-partitioning inequalities for non-atomic probability measures. Trans. Am. Math. Soc. 296, 703–725 (1986)

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  6. Hill, T.: Equipartitioning the common domain of non-atomic measures. Math. Z. 189, 415–419 (1985)

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  7. Hill, T.: Partitioning general probability measures. Ann. Prob., to appear (1987)

  8. Legut, J.: Inequalities for α-optimal partitioning of measurable space. Math. Z., to appear (1987)

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Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    Theodore P. Hill

Authors
  1. Theodore P. Hill
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Additional information

Research partly supported by NSF Grants DMS-84-01604 and DMS-86-01608

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Hill, T.P. A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures. Probab. Th. Rel. Fields 75, 143–147 (1987). https://doi.org/10.1007/BF00320087

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  • Received: 06 August 1986

  • Revised: 28 November 1986

  • Issue Date: May 1987

  • DOI: https://doi.org/10.1007/BF00320087

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Keywords

  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Total Mass
  • Statistical Theory
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