Geodesic
Quasi-invariant measures for topological dynamical systems
Izvestiya. Mathematics, Tome 8 (1974) no. 6, pp. 1287-1304 
I. P. Kornfeld. Quasi-invariant measures for topological dynamical systems. Izvestiya. Mathematics, Tome 8 (1974) no. 6, pp. 1287-1304. http://geodesic.mathdoc.fr/item/IM2_1974_8_6_a7/
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     author = {I. P. Kornfeld},
     title = {Quasi-invariant measures for topological dynamical systems},
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that for a topological dynamical system to admit an ergodic quasi-invariant measure of type III (a measure which is not equivalent to any $\sigma$-finite invariant measure) it is necessary and sufficient that this system have a recurrent point. For systems with a recurrent point, it is shown that there exist a nondenumerable number of pairwise singular ergodic quasi-invariant measures of type III.

[1] Ornstein D. S., “On invariant mesures”, Bull. Amer. Math. Soc., 66:4 (1960), 297–300 | DOI | MR | Zbl

[2] Moore C. C., “Invariant mesures on product spaces”, Proceedings of the Fifth Berkeley symposium on Math. Stat. and Probability, 1965, 447–459 | MR

[3] Keane M., “Sur les mesures quasi-ergodique des translations irrationelles”, Compt. Rend. Acad. Sci., 272:1 (1971), A54–A55 | MR

[4] Krieger W., “On quasi-invariant mesures in uniquely ergodic systems”, Inv. Math., 14 (1971), 184–196 | DOI | MR | Zbl

[5] Katsnelson I., Weiss B., “The construction of quasi-invariant mesures”, Isr. J. Math., 12:1 (1972), 1–4 | DOI | MR

[6] Billingsley P., Convergence of Probability mesures, Wiley, New York, 1968 | MR | Zbl