Convergence (topology) for $\sigma$-finite measures

Question

I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by sharing what they know --ideally both. My measure theory knowledge is quite limited and it might be some standard theory, in which case I apologize.

Let $(X,d)$ be a complete separable metric space, which is not necessarily locally compact, and $\mathcal{M}_\sigma=\{\sigma$-finite Borel measures on X$\}$ -- with the $\sigma$-fintness property of a measure $\mu$, I mean that $X$ can be covered by countable many sets of finite $\mu$-measure. I'm interested in setting a ("interesting") topology on $\mathcal{M}_\sigma$, and was wondering if there are some choices which are commonly used. I was hoping of something in a similar the direction in which one sets the weak$^*$-topology on, say, bounded Radon measures in duality with continuous bounded functions(?) for locally compact metric spaces. Some specifical questions are:

• Is $\mathcal{M}_\sigma$ the (algebraic) dual space to some subset of continuous functions (with some norm)? Or ask differently, is there a Riesz–Markov–Kakutani representation theorem in this setting?
• Are there other known common choices of topologies on $\mathcal{M}_\sigma$?
• A related question, that poped-up: is there a Riesz–Markov–Kakutani representation theorem for (non-necessarily finite) Radon measures in (non-necessarily locally compact) separable metric spaces?

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One naive idea is the following (assuming things like convergent sequences define a unique topology...). Let $\{\mu_n\}_{n\in \mathbb{N}\cup \infty}\subset \mathcal{M}_\sigma$ and denote by $\{E_j^n\}$ a countable collection of measurable sets of finite $\mu^n$ measure which cover X, for every such $n$. (The sets exists because of the $\sigma$ finiteness.) Then, we could say that the sequence $\{\mu_n\}_{n\in \mathbb{N}}$ converges to $\mu_\infty$ if for every $E_j^\infty$ we can find a sequence of $\{E_{j_n}^n\}_{n\in\mathbb{N}}$ for which the finite restricted measures $\mu_n|_{E_{j_n}^n}$ converge weakly to $\mu_\infty|_{E_j^{\infty}}$. Now this latter weak*-convergence I suppose should be better understood since they are just bounded measures. Another approach would be to provide a common countable cover of X, by elements which have finite measure for all measures in the sequence, and ask for the convergence for the measures restricted to every element of the cover. Im sure there's many problems with these approaches, which I fail to see now but maybe you could point me out where they might come up.

Anyway, the question grew big. I would aprreciate it much any helpful comments.

Answer 1

In general $\mathcal{M}_\sigma$ is not the algebraic dual of any space of continuous functions.

Suppose that $X$ has no isolated points. Let $E$ be a countable dense subset of $X$, and let $\mu$ be the counting measure on $E$, which is $\sigma$-finite because $E$ is countable. So $\mu \in \mathcal{M}_\sigma$. However, every nonempty open set $U$ contains infinitely many points of $E$, and therefore $\mu(U) = \infty$.

If $f : X \to \mathbb{R}$ is any continuous function that is not the zero function, then for sufficiently small $\epsilon>0$, the set $\{|f| > \epsilon\}$ is a nonempty open set, and it follows that $\int |f|\,d\mu \ge \epsilon \mu(|f| > \epsilon) = \infty$. So no nontrivial continuous function is integrable with respect to $\mu$, and so it does not make sense to view $\mu$ as a linear functional on any nontrivial space of continuous functions.

Answer 2

Here are some thoughts which might be of interest. As I see it, you would like, firstly, a space of functions which is in natural duality to your measures and, secondly, natural topologies which are compatible with this duality. Of course one can always use weak topologies in this situation but I assume that you would like something more substantial, as I also would. One precise formulation is that these spaces should be identified as objects of two categories of topological vector spaces which are complete in some sense and are such that there is a symmetric duality between them. Thus, if the metric space were compact, the standard Riesz duality would do, with $C(K)$ a Banach space and the space of measures a Walbroeck space. Even here one has to leave the territory of Banach spaces and consider more exotic structures. Not unexpectedly, these become more elaborate as the generality increases. Thus for the case of bounded Radon measures on a metric space, one uses the space of bounded, continuous functions and the duality between Saks and CoSaks spaces, for the space of bounded measures on a $\sigma$-algebra that of bounded, measurable functions (not equivalence classes), again in a Saks-CoSaks duality. In your case, things become still rougher—-the appropriate class of functions, as hinted at in your question, is that of those bounded measurable functions which are such that for each partition their support is contained in the union of a finite family. The corresonding structure is, as one might expect, rather complicated—-a projective limit of inductive limits of sequences of Saks spaces—-but I think that this lies in the nature of the beast.

These are just night thoughts so please regard them as comments, rather than as a solution.