Let $(X,\Sigma)$ be measurable space and $\mu_1,\mu_2,\dots$ set of finite measures on $X$ such that $\mu_i \perp \mu_j$ for $i\neq j$.
Now we can consider space of measures: $$ \mathcal{M} = \left\{ \sum_i f_i \mu_i : f_i \in L^2(X,\mu_i), \sum_i \|f_i\|^2_{L^2(X,\mu_i)} < \infty \right\} $$ This space has natural inner product $$ \left< \sum_i f_i \mu_i, \sum_i g_i \mu_i \right> = \sum_i \int_X \left| f_i g_i\right| \,d\mu_i $$
We do not have to restrict ourselves with countable combination. Consider set $\{ \mu_y : y\in[0,1]\}$ of finite measures on $X$ such that $\mu_y \perp \mu_z$ for $y\neq z$.
Than consider space of measure: $$ \mathcal{M} = \left\{ \int_0^1 f_y \mu_y \, dy : f_y \in L^2(X,\mu_y), \int_0^1 \|f_y\|^2_{L^2(X,\mu_y)} < \infty \right\} $$ The first integral defines measure as $\int_0^1 f_y \mu_y \, dy (A) =\int_0^1 \int_A f_y(x) \, d\mu_y(x)\, dy$ for $A\in \Sigma$.
Again this space can be equiped inner product too $$ \left< \int_0^1 f_y \mu_y \, dy, \int_0^1 g_y \mu_y \, dy \right> = \int_0^1 \int_X \left| f_y(x) g_y(x)\right| \,d\mu_y(x) \, dy $$
In this case there might be some technical problems with measurability of $f_y \mu_y$ in $y$ variable, I don't know but that is not the point of the question.
My questions are:
Are there any examples of spaces of measures which admit described structure?
Do Borel measures admit this structure?
- If yes - What would be the choice of measures $\mu_1,\mu_2,\dots$
- If no - Is there a subspace of Borel measures which admits this structure?
Does this structure have name? Any references?