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A remark on a theorem of Losert. (English) Zbl 0799.28008

The main result of this paper is that subject to the continuum hypothesis any Haar measure on a compact topological group with (topological) weight less or equal to \(\aleph_ 2\) admits a Baire strong lifting. This result is an extension of a theorem of Losert for products of less than \(\aleph_ 2\) Radon measures each one supported by a compact metric space and it falls under a general class of results exploiting a certain analogy between measures on products of compact metric spaces and measures on compact topological groups, a problem raised by Choksi.
The methods taken for the proof are in analogy with Losert’s procedure but the proof of the main lemma is much harder to achieve. Finally, it should be noted that the result can be extended to locally compact groups.

MSC:

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A51 Lifting theory
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References:

[1] Bourbaki, N.: Integration. Chs. 1-8. Paris: Hermann. 1959-1967.
[2] Choksi, J. R.: Recent developments arising out of Kakutani’s work on completion regularity of measures. Contemp. Math.26, 81-94 (1984). · Zbl 0538.28008
[3] Fremlin, D. H.: On two theorems of Mokobodzki. Note of 26/6/77.
[4] Kupka, J., Prikry, K.: Translation invariant Borel liftings hardly even exist. Indiana J. Math.32, 717-731 (1983). · Zbl 0516.28005 · doi:10.1512/iumj.1983.32.32047
[5] Losert, V.: A counterexample on measurable selections and strong lifting. In: K?lzow, D. (ed.) Measure Theory Oberwolfach 1979, 153-159. Berlin-Heidelberg-New York: Springer. 1980 (Lecture Notes Mathematics, 794).
[6] Montgomery, D., Zippin, L.: Topological Transformation Groups. New York: Interscience. 1955. · Zbl 0068.01904
[7] Ionescu Tulcea, A., C.: On the existence of a lifting commuting with the left translations of an arbitrary locally compact group. Proc. Fifth Berk. Symp. Math. Stat. and Prob. Vol. 2, Part 1. pp. 13-97. · Zbl 0201.49202
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