Journal article
Lebesgue measure on ℝ ∞, II
Proceedings of the American Mathematical Society, Vol.132(9), pp.2577-2591
2004
DOI: 10.1090/S0002-9939-04-07372-1
Abstract
Let ℝ ∞ be the set of real numbers, and define ℝ ∞ = Π i=1 ∞ ℝ. We construct a complete measure space (ℝ ∞, ℒ, λ) where the (σ-algebra ℒ contains the Borel subsets of ℝ ∞, and λ is a translation-invariant measure such that for any measurable rectangle R = Π i=1 ∞ R i if 0 ≤ Π i=1 ∞ m(R i) < +ℝ, then λ(R) = Π i=1 ∞ m(R i )i where m is Lebesgue measure on ℝ. The measure λ is not σ-finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure λ, we construct, via selfadjoint operators on L 2 (ℝ ∞, ℒ, λ), a "Schrödinger model" of the canonical commutation relations: [P j, P k] = [Q j, Q k] = 0, [P j, Q k] = iδ jk, 1 ≤ j, k < +∞.
Details
- Title: Subtitle
- Lebesgue measure on ℝ ∞, II
- Creators
- Richard L Baker
- Resource Type
- Journal article
- Publication Details
- Proceedings of the American Mathematical Society, Vol.132(9), pp.2577-2591
- DOI
- 10.1090/S0002-9939-04-07372-1
- ISSN
- 0002-9939
- eISSN
- 1088-6826
- Publisher
- American Mathematical Society
- Language
- English
- Date published
- 2004
- Academic Unit
- Mathematics
- Record Identifier
- 9983985934102771
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