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 ...](https://mathoverflow.net/questions/202323/is-the-space-of-all-borel-measures-on-mathbb-rn-isomorphic-to-the-tensor-pro/202351#202351)
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.



Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
In [stereotype theory](http://en.wikipedia.org/wiki/Stereotype_space) 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](http://link.springer.com/article/10.1023%2FA%3A1020929201133) (Theorem 8.4).


As a corollary (again, [here](http://link.springer.com/article/10.1023%2FA%3A1020929201133), 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).


