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Two Kinds of Invariance of Full Conditional Probabilities

Pruss A. R.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 3-4

277-283

Let G be a group acting on Ω and F a G-invariant algebra of subsets of Ω. A full conditional probability on F is a function P:F×(F∖{∅})→[0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB)=P(A|B) for all g∈G and A,B∈F, and strongly G-invariant provided that P(gA|B)=P(A|B) whenever g∈G and A∪gA⊆B. Armstrong (1989) claimed that weak and strong invariance are equivalent, but we shall show that this is false and that weak G-invariance implies strong G-invariance for every Ω, F and P as above if and only if G has no non-trivial left-orderable quotient. In particular, G=Z provides a counterexample to Armstrong's claim.

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Pruss A. R.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 2

161--168

Let Ω be a countable infinite product Ω₁N of copies of the same probability space Ω1, and let {Ξn} be the sequence of the coordinate projection functions from Ω to~Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to R, and let X_n(ω)=Ψ(Ξn(ω)). Then we can think of {X_n} as a sequence of independent but possibly nonmeasurable random variables on Ω. Let S_n=X₁+⋯+X_n. By the ordinary Strong Law of Large Numbers, we almost surely have E∗[X1]≤lim infSn/n≤lim supSn/n≤E∗[X₁], where E∗ and E∗ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sn/n in the nontrivial case where E∗[X₁]1], and obtain several negative answers. For instance, the set of points of Ω where S_n/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.