``Lebesgue measure'' on , II

Author:
Richard L. Baker

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2577-2591

MSC (2000):
Primary 28A35, 28C10, 81D05

DOI:
https://doi.org/10.1090/S0002-9939-04-07372-1

Published electronically:
April 21, 2004

MathSciNet review:
2054783

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -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 , we construct, via selfadjoint operators on , a ``Schrödinger model'' of the canonical commutation relations: , , .

**[B]**R. L. Baker,*``Lebesgue Measure'' on*, Proc. Amer. Math. Soc.**R****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

Email:
baker@math.uiowa.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07372-1

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