𝜎-finite invariant measures on infinite product spaces

Hill, David G. B.

doi:10.1090/S0002-9947-1971-0274725-1

$\sigma$ -finite invariant measures on infinite product spaces

Author: David G. B. Hill
Journal: Trans. Amer. Math. Soc. 153 (1971), 347-370
MSC: Primary 28.75
DOI: https://doi.org/10.1090/S0002-9947-1971-0274725-1
MathSciNet review: 0274725
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Abstract: A necessary and sufficient condition in terms of Hellinger integrals is established for the existence of a $\sigma$ -finite invariant measure on an infinite product space. Using this it is possible to construct a wide class of transformations on the unit interval which have no $\sigma$ -finite invariant measure equivalent to Lebesgue measure. This class includes most of the previously known examples of such transformations.

[1] Leslie K. Arnold, On 𝜎-finite invariant measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968), 85–97. MR 227362, https://doi.org/10.1007/BF01850999
[2] Antoine Brunel, Sur les mesures invariantes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 300–303 (French). MR 207957, https://doi.org/10.1007/BF00535360
[3] R. V. Chacon, A class of linear transformations, Proc. Amer. Math. Soc. 15 (1964), 560–564. MR 165071, https://doi.org/10.1090/S0002-9939-1964-0165071-7
[4] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
[5] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
[6] Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956. MR 0097489
[7] A. Ionescu Tulcea, On the category of certain classes of transformations in ergodic theory, Trans. Amer. Math. Soc. 114 (1965), 261–279. MR 179327, https://doi.org/10.1090/S0002-9947-1965-0179327-0
[8] Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, https://doi.org/10.2307/1969123
[9] Calvin C. Moore, Invariant measures on product spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 447–459. MR 0227366
[10] Donald S. Ornstein, On invariant measures, Bull. Amer. Math. Soc. 66 (1960), 297–300. MR 146350, https://doi.org/10.1090/S0002-9904-1960-10478-8