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?