Domain of Computation of a Random Field

We present a domain-theoretic analysis of the invariant measure of the one-dimensional Ising model in a random external magnetic eld. The invariant measure is obtained as a xed point of the Markov transition operator of an iterated function system with probabilities acting on the probabilistic power domain of the upper space of a closed real interval. This enables us to use the generalised Riemann integral in combination with Elton's ergodic theorem to obtain an algorithm to compute the free energy density of the system. We also develop the generalised double Riemann integral, which we use, together with a two-dimensional version of Elton's theorem, to deduce algorithms to compute the magnetisation per spin and the Edwards-Anderson parameter of the system.