Invariant relative probability measures for discrete dynamical systems created by maps

Abstract

In this paper we consider the maps which preserve a relative probability measure on a set M. We prove that a mapping g : M → M preserves a relative probability measure if and only if , for each simple function . We also prove that g preserves for each observer , if and only if g has a fixed point. We show that if there is x in M such thatfor each , then there is a mapping f : M → M such that g preserves .