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Proceedings of the American Mathematical Society

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

 

 

``Lebesgue measure'' on $ {\bf R}\sp \infty$


Author: Richard Baker
Journal: Proc. Amer. Math. Soc. 113 (1991), 1023-1029
MSC: Primary 46G12; Secondary 28A35, 28C20
MathSciNet review: 1062827
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Abstract: We construct a translation invariant Borel measure $ \lambda $ on $ {{\mathbf{R}}^\infty } = \prod _{i = 1}^\infty {\mathbf{R}}$ such that for any infinite-dimensional rectangle $ R = \prod _{i = 1}^\infty ({a_i},{b_i}), - \infty < {a_i} \leq {b_i} < + \infty $, if $ 0 \leq \prod _{i = 1}^\infty ({b_i} - {a_i}) < + \infty $, then $ \lambda (R) = \prod _{i = 1}^\infty ({b_i} - {a_i})$. Because $ {{\mathbf{R}}^\infty }$ is an infinite-dimensional locally convex topological vector space, the measure $ \lambda $ can not be $ \sigma $-finite.


References [Enhancements On Off] (What's this?)

  • [1] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • [2] Y. Yamasaki, Measures on infinite-dimensional spaces, Series in Pure Mathematics, vol. 5, World Scientific Publishing Co., Singapore, 1985. MR 999137

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1062827-X
Article copyright: © Copyright 1991 American Mathematical Society