Universal measurability of the identity mapping of a Banach space in certain topologies

If $X$ is a Banach space and $X'$ is its conjugate, then a subset $Y$ of $X'$ is called madmissible for $X$ if a) he topology $\sigma(X,Y)$ is Hausdorff, b) the identity embedding of ($X,\sigma(X,Y)$) into $X$ is universally measurable (Ref. Zh. Mat., 1975, 8B 75 8K). If $X$ is separable, then the existence of an $m$-admissible set is well known. In this note it is shown that there exist nonseparable $X$ having separable $m$-admissible sets. The properties of spaces with separable $m$-admissible sets are considered. It is proved, in particular, that a separable normalizing subset $Y$ of $X'$ is $m$-admissible for $X$ if and only if every $\sigma(X,Y)$-compact set is separable in $X$.