Ergodic properties of random measures on stationary sequences of sets

We study a class of stationary sequences having spectral representation (M([tau]nA))n[epsilon], where A is a set in a measure space (E, , [mu]), [tau] is an invertible measure-preserving transformation on (E, , [mu]), and M is a random measure on (E, , [mu]). We explore the relationship between the ergodic properties of the sequence and the properties of [tau], and construct examples with various ergodic properties using a stacking method on the half-line [0, [infinity]).