I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence:


**Theorem.** Let $\phi,\phi\_1,\phi\_2,\dotsc$ be random measures on a Polish space $S$. Then $\phi\_n \to \phi$ in distribution if and only if $\phi\_n f \to \phi f$ in distribution for every bounded continuous $f$.


Now consider this application: Let $\eta$ be a random variable, and define $\zeta\_n = \Pr[\phi\_n \in \cdot \,| \eta]$ and $\zeta = \Pr[\phi \in \cdot \,| \eta]$. I would like to show $\zeta\_n \to \zeta$ weakly almost surely. 


Naively, one might think to show that, for every bounded continuous $f$, we have 
$$\Pr [\phi\_n f \in \cdot \,|\eta] \to \Pr [\phi f \in \cdot \, | \eta] \text { weakly a.s.}$$ But this fails because of the uncountable number of bounded continuous functions. However, if I knew that there was some countable class $\mathcal C$ of bounded continuous functions such that convergence on this class implied convergence for all bounded continuous functions, then I could establish the a.s. weak convergence on this countable class and then conclude that convergence holds simultaneously, and then I can apply the Theorem above.


Is there anything that I have missed?


Are there example where one might expect to transfer an unconditional result to a conditional setting, but trouble lurks?



Moving a result from the unconditional to the conditional
There's a fairly well known result due to Varadarajan (I think in [this paper](http://www.jstor.org/stable/25048364?seq=1#page_scan_tab_contents), though I can't get through the paywall at the moment), that provides your countable class $\mathcal{C}$ you desire: If $S$ is a Polish space, then there exists a countable family $\mathcal{C}$ of continuous bounded functions of $S$ such that $\mu\_n \rightarrow \mu$ in $\mathcal{P}(S)$ if and only if $\mu\_n f\rightarrow\mu f$ for every $f \in \mathcal{C}$.


