ℕ_o vs. ℤ

Abstract

The title of this chapter calls for an explanation: Given a base space X, an endomorphism σ of X, and a prescribed weight function W on X, we saw in Section 1.2 that there is an associated measure P_x on the space of infinite paths rooted at x; see Figure 1.1 (p. 8) for an illustration. As noted, the function W determines the transition probabilities that go into the probability measure P_x as follows: For two “successive” points y and z on such a path in X, a transition is possible if σ (z)=y, and the transition probability is then W (z). If points on the path are further apart, we use a natural formula for conditional probabilities.

Every axiomatic (abstract) theory admits, as is well known, an unlimited number of concrete interpretations besides those from which it was derived. Thus we find applications in fields of science which have no relation to the concepts of random event and of probability in the precise meaning of these words. —A.N. Kolmogorov 1933