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?

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.


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