Uniform Measures on Inverse Limit Spaces

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let f : X → X where X is a compact metric space and f is continuous, onto and piecewise one-to-one and Y := lim ←− (X, f). Then starting with a measure μ1 on the Borel sets B(X), we recursively construct a sequence of probability measures {μn}n=1 on B(X) satisfying μn(A) = μn+1[f−1(A)] for each A ∈ B(X) and n ∈ N. This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y . If μ1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define an SRB measure or a metric attractor for the shift map on the inverse limit space.