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Abstract

Let $G$ be a group acting on $\Omega$ and $\scr F$ a $G$-invariant algebra of subsets of $\Omega$. A full conditional probability on $\scr F$ is a function $P:\scr F\times (\scr F\backslash \{ \varnothing \})\to[0,1]$ satisfying the obvious axioms (with only finite additivity). It is weakly $G$-invariant provided that $P(gA\,|\, gB)=P(A\,|\, B)$ for all $g\in G$ and $A,B\in \scr F$, and strongly $G$-invariant provided that $P(gA\,|\, B)=P(A\,|\, B)$ whenever $g\in G$ and $A\cup gA\subseteq B$. 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 $\Omega$, $\scr F$ and $P$ as above if and only if $G$ has no non-trivial left-orderable quotient. In particular, $G=\mathbb Z$ provides a counterexample to Armstrong's claim.