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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On genericity and complements of measure zero sets in function spaces

Author: D. Rebhuhn
Journal: Proc. Amer. Math. Soc. 68 (1978), 351-354
MSC: Primary 22A10; Secondary 58D99
MathSciNet review: 0480839
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Abstract: Generic properties of function spaces have been of particular interest in dynamical systems and singularity theory. The underlying assumption has been that the complement of a dense $ {G_\delta }$ set is sparse enough to be considered unlikely. Nevertheless, in infinite dimensional spaces, even dense $ {G_\delta }$'s may have measure zero. Since there is no one canonical measure on an infinite dimensional Fréchet space, notions of measure zero have not often been considered. Here we use a notion of Haar measure zero on abelian Polish groups due to Christensen [1]. We show that those sections of a finite dimensional vector bundle over a compact manifold whose jets are transverse to a submanifold of the jet bundle are complements of sets of Haar measure zero.

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  • [1] J. P. R. Christensen, Topology and Borel structure, American Elsevier, New York, 1974. MR 0348724 (50:1221)
  • [2] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York, 1973. MR 0341518 (49:6269)
  • [3] R. Narasimhan, Analysis on real and conplex manifolds, North-Holland, Amsterdam, 1968. MR 0251745 (40:4972)

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Keywords: Space of sections of a bundle, transversality, genericity, Polish group, Haar measure zero
Article copyright: © Copyright 1978 American Mathematical Society

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