Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well known to be a Banach space with the norm being total variation). Is there any characterization of it? We may impose extra assumptions on the underlying space if necessary.
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Well, your space of measures is isometric to $L^1(\mu)$ for some (probably very big, non-sigma-finite) measure $\mu$. So it is enough to know what is the dual of an $L^1$ space.
In the case of measures on a compact space, you are talking about the bidual of $C(K)$. This space was investigated in detail by S. Kaplan who wrote a series of long papers on it in the Transactions---easily available online. He also produced a book summarising his results. The natural extension for completely regular spaces would be the bidual of the space of bounded, continuous functions thereon, with the strict topology. This is certainly an interesting space and many of Kaplan's results carry over in suitably modified form but nobody has written this up to my knowledge.