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``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 $.

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

  • [B] R. L. Baker, ``Lebesgue Measure'' on R $^{\infty }$, Proc. Amer. Math. Soc. 113, 1023-1029 (1991). MR 92c:46051
  • [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience Publishers, Inc., New York, 1958. MR 22:8302
  • [EM] E. O. Elliott and A. P. Morse, General product measures, Trans. Amer. Math. Soc. 110, 245-283 (1964). MR 28:2178
  • [O] J. C. Oxtoby, Invariant measures in groups which are not locally compact, Trans. Amer. Math. Soc. 60, 215-237 (1946). MR 8:253d
  • [P] E. Prugovecki, Quantum Mechanics in Hilbert Space, second edition, Pure and Applied Mathematics, vol. 92, Academic Press, New York, 1981. MR 84k:81005
  • [RH] G. E. Ritter and E. Hewitt, Elliott-Morse measures and Kakutani's dichotomy theorem, Math. Zeitschrift 211, 247-263 (1992). MR 93i:28003
  • [Rg] C. A. Rogers, Hausdorff Measures, Cambridge University Press, 1970. MR 43:7576
  • [Ry] H. L. Royden, Real Analysis, Macmillan Publishing Co., Inc., New York, 1963. MR 27:1540
  • [Rd] W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987. MR 88k:00002

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Additional Information

Richard L. Baker
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242

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

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