On Sufficiency and Invariance

Abstract

Let $\mathscr{P}$ be a family of probability measures defined on a $\sigma$-field $\mathscr{A}$ on $X$ and $G$ be a group of transformations on $X$ such that $Pg^{-1} \in \mathscr{P}$ for all $P \in \mathscr{P}, g \in G$. Let $\mathscr{A}_I$ be the $\sigma$-field of $G$-invariant sets of $\mathscr{A}$ and $\mathscr{A}_{I^\ast}$ the $\sigma$-field of $\mathscr{P}$-almost $G$-invariant sets of $\mathscr{A}$. Let $\mathscr{A}_S$ be a sufficient $\sigma$-field for $\mathscr{P} \mid \mathscr{A}$. Hall, Wijsman and Ghosh proved that $\mathscr{A}_S \cap \mathscr{A}_I$ is sufficient for $\mathscr{P} \mid \mathscr{A}_I$ if $g\mathscr{A}_S = \mathscr{A}_S$ for each $g \in G$ and $\mathscr{A}_S \cap \mathscr{A}_I \sim \mathscr{A}_S \cap \mathscr{A}_{I^\ast}(\mathscr{P})$. They posed the question whether the first condition alone suffices to prove this result. An example shows that the answer is no. For dominated families we show that $\mathscr{A}_S \cap \mathscr{A}_{I^\ast}$ is always sufficient for $\mathscr{P} \mid \mathscr{A}_{I^\ast}$, a result which is not true any more for undominated families.