Probability Theory and Related Fields

, Volume 100, Issue 2, pp 175–189

Metric marginal problems for set-valued or non-measurable variables

  • R. M. Dudley

DOI: 10.1007/BF01199264

Cite this article as:
Dudley, R.M. Probab. Th. Rel. Fields (1994) 100: 175. doi:10.1007/BF01199264


In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.

Mathematics Subject Classifications

60B05 60B10 

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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