Mixed ratio limit theorems for Markov processes

Let P be a conservative and ergodic Markov operator on L₁(X, Σ, m). We give a sufficient condition for the existence of a decomposition A_f ↑X such that for 0≦f, g ∈L_∞ (A_j) and any two probability measures μ and ν weaker than m(Formula presented.), where λ is the σ-finite invariant measure (which necessarily exists). Processes recurrent in the sense of Harris are shown to have this decomposition, and an analytic proof of the convergence of(Formula presented.) is deduced for such processes.