I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence.
$\mathfrak{F}([0,1])$ is the set of distribution functions over the set $[0,1]$.
$f$ and $f^n$ are functions that to each $x\in [0,1]$ chose a distribution function on $\mathfrak{F}([0,1])$ $f, f^n: [0,1] \rightarrow \mathfrak{F}([0,1])$. Is there a good metric to evaluate the distance of $f$ and $f^n$? what kind of convergence is better suited for this, what is $f^n \rightarrow f$? Pointwise convergence, $f^n(x) \rightarrow f(x) $ and supremum metric?
If $h:[0,1]^2 \rightarrow [0,1]$ is continuous and $ g \in \mathfrak{F}([0,1])$, is $E_h=\int_{(x,y) \in [0,1]^2} h(x,y) \partial (g \times f)$ well defined? When can I say that $E^n_h=\int_{(x,y) \in [0,1]^2} h(x,y) \partial (g^n \times f^n)$ converges to $E_h$. If $f^n$ was just a distribution from $\mathfrak{F}([0,1])$, there was no problem, but with $f^n$ defined as it is can I apply a weak convergence argument here?