Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let ${ℬ}_w$ be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that Pis quasi-compact on $({ℬ}_w,\parallel ·{\parallel }_w)$ if and only if, for all $f\in {ℬ}_w$, ${\left(\frac{1}{n}{\sum }_{k=1}^n{P}^{k}f\right)}_n$ contains a subsequence converging in ${ℬ}_w$ toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of...