Two Kinds of Invariance of Full Conditional Probabilities

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.