On weakly mixing Markov operators and non-singular transformations

Let P be a Markov operator on L_∞(X, Σ, m). Theorem 1: (i) P is weakly mixing ⇔ (ii) For every f∈L_∞ there is a sequence {n_t} of density 1 such that all w^*-cluster points of {Mathematical expression} are constants ⇔ (iii) For every f∈L_∞ there is a {k_j} with {Mathematical expression}w^*-convergent to a constant. Theorem 2: If P is induced by a non-singular transformation θ, P is weakly mixing ⇔ For every AεΣ, {θ^-n(A)} has a remotely trivial subsequence. The existence of a finite invariant measure is not required in these results.