# Dual space of the space of finite measures

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.

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Dunford and Schwartz, Linear Operators I doesn't contain a description of that space. In a footnote on p.374 they remark that no satisfactory descriptions were known at the time of writing. This suggests that there is no easy characterization as you're asking for. –  t.b. Oct 22 '11 at 17:41
@t.b.: Thanks for sharing. Perhaps the one GEdgar gives below is one the "various sorts of representations" they mentioned in the footnote? –  Syang Chen Oct 23 '11 at 3:34
yes, I think so. It is unsatisfactory in that you can't say what you really get. Something humungous, in any case. –  t.b. Oct 23 '11 at 3:42

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.
Take a maximal family $\mathcal A$ of mutually singular probability measures. (Use Zorn's Lemma.) The space of measures is isometrically the $l_1$-sum of $L^1(\nu)$ as $\nu$ ranges over the family $\mathcal A$. –  GEdgar Oct 22 '11 at 22:13
So there is no canonical representation for it? What's dual of $L^1$ when the space is not $\sigma$-finite? –  Syang Chen Oct 23 '11 at 4:03
@Xianghong: Note that you have an $l_1$-sum of $L^1(\nu)$ where $\nu$ are probabilities. Its dual space is the $l_\infty$-product of the corresponding $L^\infty(\nu)$-spaces. –  t.b. Oct 23 '11 at 10:02
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.