Let me apologise in advance for the incomplete notation, however I am trying to understand this matter and I am not sure how to go about that; I did not managed to find any paper on the subject.
I want to approximate a function $V:\mathbb{R}\times\mathcal{M}\to \mathbb{R}$ with a linear combination of basis functions $\phi_{k}:\mathbb{R}\times\mathcal{M}\to \mathbb{R}$, $k=1,\dots, K$; where $\mathcal{M}$ is a space of measures on $\mathbb{R}$.
Formally, denoting $\hat{V}$ such approximation, we have $\hat{V}(x,\mu)=\alpha\cdot \phi(x,\mu)$; where we compute the "regression coefficients" as follow:
$\alpha=argmin_{\alpha\in\mathbb{R}^K}\Big\{\Big\|V(x,\mu)-\alpha\cdot \phi(x,\mu)\Big\|^2_{L^2{?}}\Big\} $.
At this point I am kind of lost; I would like to use a "Lebegue" measure on $\mathbb{R}\times\mathcal{M}$ in order to give a meaning to the norm in the equation above. I know that some restrictive assumptions are needed in order to say something about the problem, but I would like to have a general picture of this problem.
In particular I believe I finally understood that in the following specific case things should go in this way:
Assume $\mathcal{M}$ is the space of Gaussian measures. It can be proved that $\mathcal{M}$ is isomorphic to $\mathbb{R}\times\mathbb{R}^+\setminus\{0\}$ and therefore, denoting by $m$ and $\sigma^2$ the mean and the variance:
$\alpha = argmin_{\alpha\in\mathbb{R}^K}\Big\{\iiint_{\mathbb{R}\times\mathcal{M}}\big(V(x,\mu)-\alpha\cdot \phi(x,\mu)\big)^2 $d$\xi\Big\} $
$\quad=argmin_{\alpha\in\mathbb{R}^K}\Big\{\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}^+\setminus\{0\}}\big(V(x,m,\sigma)-\alpha\cdot \phi(x,m,\sigma)\big)^2 $d$\sigma $d$m$d$x\Big\} $.
This is as far as I have got (assuming I am not completely wrong); now I would like to know few things, for example who should be the measure $\xi$ to use over $\mathbb{R}\times\mathcal{M}$? I think that if $\mathcal{M}$ has some parametrization on a finite dimensional space then the approach above still applies, what if however it does not exists a finite dimensional representation of $\mathcal{M}$? Is there any hope of finding $\alpha$ explicitly?
Thanks in advance!
P.S. I am mostly thinking of probability measures really, not about general measures.