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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

``Lebesgue measure'' on $\mathbb{R}^{\infty }$, II


Author: Richard L. Baker
Journal: Proc. Amer. Math. Soc. 132 (2004), 2577-2591
MSC (2000): Primary 28A35, 28C10, 81D05
Published electronically: April 21, 2004
MathSciNet review: 2054783
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Abstract: Let $\mathbb{R}$ be the set of real numbers, and define $ \mathbb{R}^{\infty }=\prod \limits ^{\infty }_{i=1}\mathbb{R}$. We construct a complete measure space $(\mathbb{R}^{\infty },\mathcal{L},\lambda )$ where the $\sigma $-algebra $\mathcal{L}$ contains the Borel subsets of $\mathbb{R}^{\infty }$, and $\lambda $ is a translation-invariant measure such that for any measurable rectangle $R=\prod \limits ^{\infty }_{i=1}R_{i}$, if $0\le \prod \limits ^{\infty }_{i=1}m(R_{i})<+\infty $, then $\lambda (R)=\prod \limits ^{\infty }_{i=1}m(R_{i})$, where $m$ is Lebesgue measure on $\mathbb{R}$. The measure $\lambda $ is not $\sigma $-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 $\lambda $, we construct, via selfadjoint operators on $L_{2}(\mathbb{R}^{\infty },\mathcal{L},\lambda )$, a ``Schrödinger model'' of the canonical commutation relations: $[P_{j},P_{k}]=[Q_{j},Q_{k}]=0$, $[P_{j},Q_{k}]=i\delta _{jk}$, $1\le j,k<+\infty $.


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Additional Information

Richard L. Baker
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: baker@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07372-1
PII: S 0002-9939(04)07372-1
Keywords: Canonical commutation relations, Elliott-Morse measures, Fubini theorem, Fubini-Jensen theorem, infinite-dimensional Lebesgue measure, invariant measures, Schr\"{o}dinger model
Received by editor(s): August 16, 1994
Received by editor(s) in revised form: March 21, 2003
Published electronically: April 21, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society