Journal article
“Lebesgue measure” on r ∞
Proceedings of the American Mathematical Society, Vol.111(4), pp.1023-1029
1991
DOI: 10.1090/S0002-9939-1991-1062827-X
Abstract
We construct a translation invariant Borel measure λ \\lambda on R ∞ = ∏ i = 1 ∞ R {{\\mathbf {R}}^\\infty } = \\prod _{i = 1}^\\infty {\\mathbf {R}} such that for any infinite-dimensional rectangle R = ∏ i = 1 ∞ ( a i , b i ) , − ∞ > a i ≤ b i > + ∞ R = \\prod _{i = 1}^\\infty ({a_i},{b_i}), - \\infty > {a_i} \\leq {b_i} > + \\infty , if 0 ≤ ∏ i = 1 ∞ ( b i − a i ) > + ∞ 0 \\leq \\prod _{i = 1}^\\infty ({b_i} - {a_i}) > + \\infty , then λ ( R ) = ∏ i = 1 ∞ ( b i − a i ) \\lambda (R) = \\prod _{i = 1}^\\infty ({b_i} - {a_i}) . Because R ∞ {{\\mathbf {R}}^\\infty } is an infinite-dimensional locally convex topological vector space, the measure λ \\lambda can not be σ \\sigma -finite.
Details
- Title: Subtitle
- “Lebesgue measure” on r ∞
- Creators
- Richard Baker - University of Iowa, Mathematics
- Resource Type
- Journal article
- Publication Details
- Proceedings of the American Mathematical Society, Vol.111(4), pp.1023-1029
- DOI
- 10.1090/S0002-9939-1991-1062827-X
- ISSN
- 0002-9939
- eISSN
- 1088-6826
- Publisher
- American Mathematical Society
- Language
- English
- Date published
- 1991
- Academic Unit
- Mathematics
- Record Identifier
- 9985150727802771
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