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Lower layers in R 2 and convex sets in R 3 are not GB classes

Part of the Lecture Notes in Mathematics book series (LNM,volume 709)

Abstract

In R 2 let <u,v> < <x,y> iff u<x and v<y. A lower layer is a set A √ R 2 such that if <u,v> < <x,y> ε A then <u,v> ε A. Let λ be Lebesgue measure on a bounded, open, non-empty set in R 2. Let W be the Gaussian process indexed by Borel sets with EW(A)=0 and EW(A)W(B)=λ(A ∩ B) (white noise).

It is proved that W has almost all sample functions unbounded on the collection LL of all lower layers, i.e. LL is not "GB." Likewise, in R 3 the collection of all convex subsets of the unit ball is not GB.

Keywords

  • Lebesgue Measure
  • Lower Layer
  • Unit Ball
  • Gaussian Process
  • Sample Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This research was partially supported by National Science Foundation Grant MCS76-07211 A01.

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References

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© 1979 Springer-Verlag

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Dudley, R.M. (1979). Lower layers in R 2 and convex sets in R 3 are not GB classes. In: Beck, A. (eds) Probability in Banach Spaces II. Lecture Notes in Mathematics, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071951

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  • DOI: https://doi.org/10.1007/BFb0071951

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09242-1

  • Online ISBN: 978-3-540-35341-6

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