Invariant relative probability measures for discrete dynamical systems created by maps

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 m(mu)(f) if and only if integral X ogdm(mu)(f) = integral X dm(mu)(f), for each simple function X : M ->Pi R-i is an element of I. We also prove that g preserves m(mu)(f) for each observer mu:M -> Pi(i is an element of I)[0,1], if and only if g has a fixed point. We show that if there is x in M such that lim sup(n ->infinity) Sigma(n)(i=1) a(i+1),mu(g(i)(x))/n = lim sup(n ->infinity )Sigma(n)(i=1) a(i),mu(g(i)(x))/n for each {a(i)}(i=1)(infinity )subset of {0, 1}, then there is a mapping f : M -> M such that g preserves m(mu)(f).