Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compactopen topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong projective limit of the spaces $C(K)$, where $K$ is a compact subset of $K$?

2$\begingroup$ Related questions: mathoverflow.net/questions/145215/… and mathoverflow.net/questions/105147/… $\endgroup$ – András Bátkai Jun 20 '18 at 11:09

$\begingroup$ No. By the references mentioned by András Bátkai, $\mathcal{C}(X)^*$ can be identified with the space of (Radon) measures on $X$ with compact support. This space maps by restriction to each $\mathcal{C}(K)^*$, hence to their projective limit, but the latter map is not surjective: the family of Lebesgue measures on compacts of $\mathbb{R}$ does not come from a compactly supported measure on $\mathbb{R}$. $\endgroup$ – abx Jun 20 '18 at 13:44

$\begingroup$ @abx, but if you take such a measure $\mu$ (supported on the whole real line), how do you guarantee that $\int f d \mu$ is finite for every (unbounded) continuous function? $\endgroup$ – user125821 Jun 20 '18 at 15:08

$\begingroup$ I don't. This is why the answer is negative. $\endgroup$ – abx Jun 20 '18 at 16:59

$\begingroup$ @abx, sorry I don't understand. What are Lebesgue measures for you? $\endgroup$ – Tomasz Kania Jun 20 '18 at 17:41