Invariance, almost-invariance and sufficiency

Abstract

This paper is concerned with the relationship between sufficiency, invariance and conditional indipendence. Given a sufficient [sigma]-field, two results are obtained about independence and conditional independence of it and  the almost invariant [sigma]-field. Moreover, it includes a rigorous proof of a stronger result  than a Stein theorem about sufficiency and invariance in its version for statistics and in the discrete case. Somewhat of a strange condition is used to prove this theorem in its original form; several results are given here to clarify it.