Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Question

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

This question is strongly linked to

is the space of all borel measures on $\mathbb{R}^n$ isomorphic to the tensor ... where my answer was (too) quickly accepted and turned out to be more a plan than a a satisfying answer.

Let me elaborate a bit, let $X,Y$ be two locally compact spaces

We consider the corresponding spaces of measures $\mathcal{M}(X),\mathcal{M}(Y),\mathcal{M}(X\times Y)$ and the canonical map $$ can : \mathcal{M}(X)\otimes \mathcal{M}(Y)\rightarrow \mathcal{M}(X\times Y) $$ defined by $$ can(\mu\otimes\nu)(f\otimes g)=\mu(f)\nu(g) $$

As I am not an expert in measure theory (and after having searched and discussed a bit), I ask the following questions whose clarification might be enlighting to read.

Q1) Is $can$ always into ? (generalization/discussion for measurable spaces ?)

Q2) For which complete TVS topology on $\mathcal{M}(X\times Y)$ is the image of $can$ dense in $\mathcal{M}(X\times Y)$ ? (well-known or hand-made answers are accepted).

Q3) Same questions for bounded measures as $$ can(\mathcal{M}^{1}(X)\otimes \mathcal{M}^{1}(Y))\subset \mathcal{M}^{1}(X\times Y) $$

Please give references and/or proofs.

Answer 1

In stereotype theory there is an isomorphism of stereotype spaces (or, what is the same here, an isomorphism of locally convex spaces) $$ {\mathcal C}^\star(X)\circledast {\mathcal C}^\star(Y)\cong{\mathcal C}^\star(X\times Y), $$ where $X$ and $Y$ are arbitrary paracompact locally compact spaces, ${\mathcal C}(Z)$ the algebra of continuous functions on $Z$ ($Z=X$, or $Y$, or $X\times Y$) with the topology of uniform convergence on compact sets in $Z$, and ${\mathcal C}^\star(Z)$ the stereotype dual space (here the space of all compactly supported measures on $Z$ with the topology of uniform convergence on compact sets in ${\mathcal C}(Z)$). This is proved here (Theorem 8.4).

As a corollary (again, here, Proposition 7.2), the map
$$ {\mathcal C}^\star(X)\otimes {\mathcal C}^\star(Y)\to {\mathcal C}^\star(X\times Y) $$ is injective and has dense image in ${\mathcal C}^\star(X\times Y)$.

If by ${\mathcal M}(Z)$ you mean the space of all measures, then for compact $Z$ it coincides as a set with ${\mathcal C}^\star(Z)$, $$ {\mathcal M}(Z)={\mathcal C}^\star(Z) $$ and this endows ${\mathcal M}(Z)$ with natural topology satisfying the properties that you want (I mean, always Q2 and, if "into" means injectivity, then Q1).

Answer 2

Q1 has a positive answer and your question about density has a simple positive answer in both cases by using appropriate weak topologies. One can give your queries more content by looking for COMPLETE structures. Again the responses are both positive as I thought I had pointed out in my response to the question you mention. In the case of bounded measures, then we have the dual of a Banach space (of continuous functions which vanish at infinity) and you can use the bounded weak star topology which is the finest locally convex (or, indeed, any) topology which agrees with the weak star topology on bounded sets.

In the unbounded case it is standard that the space of measures is, in a natural way, the projective limit (in the category of vector spaces) of the spaces $M(K)$ as $K$ ranges over the compacta. One then regards it a lcs by taking the projective limit of the above complete topologies.