Given a Banach space $X$, we say that a subset $Y\subseteq X^*$ is Orlicz-Thomas (OT), if for every $\sigma$-algebra $\Sigma$ and every map $\mu\colon \Sigma \to X$, the countable additivity of all the compositions $\varphi\circ \mu$ for $\varphi\in Y$ implies the countable additivity of $\mu$. The classical Orlicz-Pettis theorem says that $X^*$ is OT for any $X$. If $X$ does not contain an isomorphic copy of $\ell_\infty$, then by a result of Diestel and Faires $Y\subseteq X^*$ is OT if and only if it is total.

During the talk I will review recent progress on description of OT subsets of $\ell_\infty^*$. This will include new examples as well as sufficient and necessary conditions for being OT.