We show that for a bounded subset $A$ of the $L_{1}(\unicode[STIX]{x1D707})$ space with finite measure $\unicode[STIX]{x1D707}$, the measure of weak noncompactness of $A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space $L_{1}$ has a subsequence whose arithmetic means are norm convergent.

Let $C^*$-algebras $A$ and $B$ be Morita equivalent and let $X$ be an $A$–$B$-imprimitivity bimodule. Suppose that $A$ or $B$ is unital. It is shown that $X$ has the weak Banach–Saks property if and only if it has the uniform weak Banach–Saks property. Thus, we conclude that $A$ or $B$ has the weak Banach–Saks property if and only if $X$ does so. Furthermore, when $C^*$-algebras $A$ and $B$ are unital, it is shown that $X$ has the Banach–Saks property if and only if it is finite dimensional.