Strong liftings with application to measurable cross sections in locally compact groups
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There are two principal theorems. The adjustment theorem asserts that a lifting may be changed on a set of measure zero so as to become slightly stronger. In conjunction with the standard lifting theorem, it yields generalizations (with shorter proofs) of a number of known results in the theory of strong liftings. It also inspires a characterization of strong liftings, when the measure is regular, by the fact that they induce upon every open set an artificial “closure” of that set which differs from it by a set of measure zero. The projection theorem asserts that, in the presence of a strict disintegration, a strong lifting may be transferred or “projected” from one topological measure space onto another. In conjunction with Losert's example, it yields regular Borel, measures, carried on compact Hausdorff spaces of arbitrarily large weight, which everywhere fail to have the strong lifting property. It also provides the final link needed to obtained, with no separability assumptions, a measurable cross section (or right inverse) for the canonical map Ω:G→G/H, whereG is an arbitrary locally compact group, and whereH is an arbitrary closed subgroup ofG.
KeywordsMeasure Space Compact Group Inverse Image Measurable Cross Section Compact Hausdorff Space
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