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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-9,
author = {Alexander R. Pruss},
title = {Two Kinds of Invariance of Full Conditional Probabilities},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {61},
year = {2013},
pages = {277-283},
zbl = {1311.60011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-9}
}
Alexander R. Pruss. Two Kinds of Invariance of Full Conditional Probabilities. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) pp. 277-283. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-9/