Let $X$ be a completely regular Hausdorff space, $C\_b(X)$ the Banach space of bounded continuous function and $M(X) \subseteq M(\beta X) = C(\beta X)' = C\_b(X)'$ the spaces of Radon measures on $X$. For the strong topologies it then holds


$$ \beta(C\_b(X), M(X)) \subseteq \beta(C\_b(X), M(\beta X)) = \lVert \cdot \rVert\textrm{-topology}. $$


1. Is it true that $\beta(C\_b(X), M(X)) = \lVert \cdot \rVert\textrm{-topology}$ (or at least if $X$ is locally compact)? One has to show that every bounded set in $M(\beta X)$ is contained in the closure (or bipolar) of a bounded set of $M(X)$.
2. If not, what is the $\beta(C\_b(X), M(X))$-dual in terms of measures?


(This is a question I posted in [MSE](https://math.stackexchange.com/questions/3461336/identification-of-a-strong-topology) a few days ago, which showed low interest there.)



Identification of some strong topology
The set of all Dirac measures is bounded and its polar in the space of bounded continuous functions is the unit ball of the supremum-norm -- do I miss something?


