Consider the space $C_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C_c(\mathbb{R}) = \bigcup_{n \in \mathbb{N}} C_c(\mathbb{R}, K_n)$ where $K_n$ is some compact exhaustion of $\mathbb{R}$ by compact sets and $C_c(\mathbb{R}, K_n)$ is the Banach space with uniform norm. In particular, $C_c(\mathbb{R})$ is an LB-space. Let $M := M(\mathbb{R})$ denote the topological dual $C_c'(\mathbb{R})$. Then $M$ can be identified with the space of all Radon measures (bounded and unbounded, e.g. the Lebesgue measure belongs to $M$) - for details, see Bourbaki: Integration, Chapter III. Every measure $\mu \in M$ has the Jordan decomposition $\mu = \mu^+ - \mu^-$ into two positive measures $\mu^+$ and $\mu^-$. Let $M^+ \subseteq M$ denote the subset of all positive measures. Then we have a surjective map $s : M^+ \times M^+ \to M$, $(\mu, \nu) \mapsto \mu - \nu$.

One can equip $M$ with the weak-* topology $\tau_v$ which Bourbaki calls the vague topology.

Some properties of $(M, \tau_v)$:

- $M$ is by duality the projective limit of the weak-* duals $C_c'(\mathbb{R}, K_n)$.
- $M$ is Hausdorff locally convex and in particular, addition and scalar multiplication are continuous
- $M$ is not first-countable and thus not metrizable
- $M$ is quasi-complete but not complete
- The map $s$ is continuous ($M^+$ carries the subspace topology (which is Polish))
- $M$ is Souslin (as the image of the Polish space $M^+ \times M^+$ under the continuous map $s$)
- $M$ is separable (since Souslin)

I don't know of the following:

- Is $M$ sequential or even Fréchet-Urysohn? (This is a specialization of the question here.)
- Is $M$ Lusin? (For an uncertain proof idea see below.)

Bourbaki studies also other topologies like compact convergence or strictly compact convergence. But one has also the map $s$ from above and can equip $M$ with the final topology $\tau_f$ induced by $s$ which is then finer than $\tau_v$ and turns $M$ into a quotient space. In particular, $(M, \tau_f)$ is sequential. Does anyone know whether this topology on $M$ was already studied before and has some nice properties (or missing some good properties)? In particular, is $\tau_f$ a vector space topology (addition and scalar multiplication continuous)?

Proof for $(M, \tau_v)$ is Lusin: I'm trying to transfer the ideas of Trèves, "Topological Vector Spaces", p. 556 (in particular Proposition A.9) which is based on the Borel Graph Theorem. $(M, \tau_v)$ is the projective limit of the countable family $C_c'(\mathbb{R}, K_n)$ equipped with the weak-* topology $\tau_{w*}$. The space $C_c'(\mathbb{R}, K_n)$ equipped with strong norm topology is Banach but it is not separable, thus not Polish and thus we can't directly say whether $(C_c'(\mathbb{R}, K_n), \tau_{w*})$ is Lusin. Now fix $n$ and set $E := C_c(\mathbb{R}, K_n)$. Since $E$ is separable Banach we have a countable basis $U_k$ of open nbhds. of $0$ and the dual $E'$ is the union of all the polars $U_k^0$. Each $U_k^0$ is weakly compact (Banach-Alaoglu) and thus weakly closed equicontinuous. By Trèves Exc. 32.9 it follows that $U_k^0$ equipped with the weak-* topology is metrizable compact, thus Polish and therefore Lusin. A countable union of Lusin subsets of a Hausdorff space is Lusin and thus $E'$ with the weak-* topology is Lusin. In other words, $(C_c'(\mathbb{R}, K_n), \tau_{w*})$ is Lusin. Moreover, the product of the countable family $(C_c'(\mathbb{R}, K_n), \tau_{w*})$ of Lusin spaces is Lusin and the projective limit $M$ of all the $(C_c'(\mathbb{R}, K_n), \tau_{w*})$ is a closed subset of the product thus a Borel subset of the product and thus Lusin.

The only transfer of the proof is that I replaced "Souslin space" by "Lusin space". So is there some major gap which I did oversee?