Q1 has a positive answer and your question about density has a simple positive answer in both cases by using appropriate weak topologies. One can give your queries more content by looking for COMPLETE structures. Again the respnses are both positive as I thought I had pointed out in my response to the question you mention. In the case of bounded measures, then we have the dual of a Banach space (of continuous functions which vanish at infinity) and you can use the bounded weak star topology which is the finest locally convex (or, indeed, any) topology which agrees with the weak star topology on bounded sets.

In the unbounded case it is standard that the space of measures is, in a natural way, the projective limit (in the cetegory Of vector spaces) of the spaces $M(K)$ as $K$ ranges over the compacta. One then regards it a lcs by taking the projective limit of the above complete topologies.

Q1 has a positive answer and your question about density has a simple positive answer in both cases by using appropriate weak topologies. One can give your queries more content by looking for COMPLETE structures. Again the responses are both positive as I thought I had pointed out in my response to the question you mention. In the case of bounded measures, then we have the dual of a Banach space (of continuous functions which vanish at infinity) and you can use the bounded weak star topology which is the finest locally convex (or, indeed, any) topology which agrees with the weak star topology on bounded sets.

In the unbounded case it is standard that the space of measures is, in a natural way, the projective limit (in the category of vector spaces) of the spaces $M(K)$ as $K$ ranges over the compacta. One then regards it a lcs by taking the projective limit of the above complete topologies.