IX: 22, 437-442, LNM 465 (1975)
MOKOBODZKI, Gabriel
Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (Measure theory)
This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see Vector and Operator Valued Measures, Academic Press 1973
Comment: The same proof leads to a slightly stronger and useful result (Meyer 2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set
Keywords: Continuum axiom, Lifting theorems, Negligible sets
Nature: Original
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