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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, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory; North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). MR 0348724
  • [2] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518
  • [3] Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR 0251745

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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