  In 1989 Kusuoka started the study of probability measures on the shift space
that are defined with the help of products of matrices. In particular, he
derived a sufficient condition for the ergodicity of such measures, which have
since been referred to as Kusuoka measures. We observe that repeated
measurements of a unitarily evolving quantum system generate a Kusuoka measure
on the space of sequences of measurement outcomes. We show that if the
measurement consists of scaled projections, then Kusuoka's sufficient
ergodicity condition can be significantly simplified. We then prove that this
condition is also necessary for ergodicity if the measurement consists of
uniformly scaled rank-1 projections (i.e., it is a rank-1 POVM), or of exactly
two projections, one of which is rank-1. For the latter class of measurements
we also show that the Kusuoka measure is reversible in the sense that every
string of outcomes has the same probability of being emitted by the system as
its reverse.
