VII: 19, 198-204, LNM 321 (1973)
MEYER, Paul-André
Limites médiales d'après Mokobodzki (
Measure theory,
Functional analysis)
Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see
304). The same problem is now solved for weak convergences, using a special kind of Banach limits
Comment: The paper contains a few annoying misprints, in particular p.199 line 9
s.c;s. should be deleted and line 17
atomique should be
absolument continu. For a misprint-free version see Dellacherie-Meyer,
Probabiliés et Potentiel, Volume C, Chapter X,
55--57
Keywords: Continuum axiom,
Weak convergence of r.v.'s,
Medial limitNature: Exposition
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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 setsNature: Original Retrieve article from Numdam
