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 $\mathcal {B}_{w}$ be the space of measurable functions on E satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that P is quasi-compact on $(\mathcal {B}_{w},\|\cdot\|_{w})$ if and only if, for all $f\in \mathcal {B}_{w}$ , $(\frac{1}{n}\sum_{k=1}^{n}P^{k}f)_{n}$ contains a subsequence converging in $\mathcal {B}_{w}$ to Πf=∑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 P-invariant functions in $\mathcal {B}_{w}$ is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.