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We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory 17 177-86). We extend Ulam's construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogousdoi:10.1088/0951-7715/12/4/318 fatcat:7jswvzxezbaornxraoks64xhai