# Reference request: sequential weak* topology on the space of signed Radon measures

Consider the space $$\mathcal{M}_{loc} (\mathbb{R}^d)$$ of locally finite signed Radon measures, equipped with the weak* topology in duality with $$C_b (\mathbb{R}^d)$$. It is known that this is space is not metrizable, nor first countable (although I believe it is a Souslin space?).

On the other hand, in practice one often works not with the weak* topology directly, but with weak* convergence. So it would be interesting to look at the sequential weak* topology on $$\mathcal{M}_{loc} (\mathbb{R}^d)$$ (that is, a set is declared to be closed provided it is sequentially weak* closed).

My question is, is this topology something that has already been studied in the literature? The only thing I can find is this MO question from several months ago.

EDIT: As user @memorial points out, the original statement of this question contains a basic error, since there is not actually a duality pairing between $$C_b (\mathbb{R}^d)$$ and $$\mathcal{M}_{loc} (\mathbb{R}^d)$$, rather, between $$C_b (\mathbb{R}^d)$$ and $$\mathcal{M} (\mathbb{R}^d)$$, and between $$C_c (\mathbb{R}^d)$$ and $$\mathcal{M}_{loc} (\mathbb{R}^d)$$. Still, the question remains, since (as far as I know) the resulting weak* topologies are not themselves sequential.

• It is not clear to me that the set of sequentially weak*-closed subsets defines a topology. Do you have a reference? Jul 11 at 8:28
• Regarding the dual space of $C_b(X)$, please see mathoverflow.net/questions/70611/… Jul 11 at 12:27
• @DieterKadelka this is discussed in Chapter 1 of Buttazzo's Semicontinuity, Relaxation and Integral Represenentation in the Calculus of Variations. Unfortunately the reference therein for the fact that the sequentially closed sets always form a topology is to a paper by Dolcher which is in Italian. Jul 11 at 20:16
• @DieterKadelka actually the Wikipedia page en.wikipedia.org/wiki/… contains a proof that the sequentially open sets form a topology. Jul 12 at 13:19

This is not an answer since I cannot give the references you are requesting for reasons that will soon be apparent but am not entitled to comment. Firstly, there is something fundamentally wrong with the framework of your question--there is no duality between spaces of bounded, continuous functions on euclidean space and locally bounded measures thereon. (Try integrating the sine function on the line with respect to Lebesgue measure). What one does have are natural dualities

a) between the bounded, continuous functions and the finite Radon measures;

and

b) between the space of continuous functions with compact support and the locally finite measures.

Note that it has long been known that in case a) one cannot use the norm topology for this duality since it leads to a much larger dual space. This was remedied in the 50´s by R.C. Buck who introduced a complete l.c. topology (the strict topology) on the bounded, continuous functions which has the bounded Radon measures as dual. He worked in the context of functions and measures on locally compact spaces but this was soon extended to that of general completely regular spaces by various authors.

In b) one uses the natural l.c. topology on the functions of compact support as a so-called strict $$LF$$-space (Dieudonne and Schwartz). This was the basis of the Bourbaki approach to measure spaces, defining them as the dual spaces of suitable locally convex spaces of continuous functions (a precursor to Schwartz´ treatment of distributions).

With regard to your question, let me start with the case of functions and measures on a compact subset. In this case there is a natural l.c. topology on the space of measures which is complete and such that the associated convergent sequences are the ones in your question. This is the finest topology which agrees with the weak star one on bounded sets. Despite this description, it is l.c. I imagine that it coincides with the one you describe but I haven´t sat down to check this. This applies to any compact metrisable space, by the way.

At the moments, due to covid, I am not in a position to give precise references but the relevant themes are the Banach-Dieudonne theorem and the bounded weak star topology. There is a reasonable article on the latter in Wikipedia but it is, rather curiously, in German (with no English version). Memory suggests that they are discussed in Koethe´s monograph and Schaefer´s standard text.

In the non-compact case, the natural topology on the locally bounded measures would be a corresponding projective limit of such space. This is then isomorphic to a closed subspace of a product of spaces of the type described above which means that it will inherit many properties from those of the compact case. It is even a complemented subspace (in the l.c. sense) which makes the situation even more tractable. I haven´t thought about whether it can be described as in your query. I am not aware of any source which deals with this directly but it is standard manipulation of inductive and projective limits of locally convex spaces and their duals.

• Thank you for spotting this serious issue, and for your long comment! It is certainly food for thought. Jul 11 at 20:25
• if you know of a reference in the literature to "natural l.c. topology on the space of measures which is complete and such that the associated convergent sequences are the ones in your question" and add it to your answer, I will accept it. Jul 12 at 13:21