Let $(\Omega,\mathcal{F})$ be measurable space and $\mathcal{P}$ be a family of probability measures on $(\Omega,\mathcal{F})$. A "typical" statistical problem is to show that
$$\tag{1}\label{1} \limsup_{n\to \infty}\sup_{P\in \mathcal{P}}P\{A_n\}\le \alpha. $$
for some constant $\alpha\in(0,1)$ and a sequence of sets $\{A_n\}_{n\ge 1}$. Violation of \eqref{1} means that there are $\epsilon>0$ and a subsequence $n_k$ s.t. $P_{n_k}\{A_{n_k}\}\ge\alpha+\epsilon$.
Now let $\mathcal{G}\subset\mathcal{F}$. Is it possible to state a conditional version of \eqref{1} given $\mathcal{G}$? If $\mathcal{P}$ is a singleton, i.e. $\mathcal{P}=\{P_0\}$, then it's trivial: $$\tag{2}\label{2} \limsup_{n\to \infty}P_0\{A_n\mid \mathcal{G}\}\le \alpha \quad P_0\text{-a.s.} $$
and violation of \eqref{2} means that on a set having positive $P_0$-probability $P_0\{A_n\mid \mathcal{G}\}(\omega)$ exceeds $\alpha$ infinitely often.
Specifically in my application $\mathcal{G}$ is generated by a sequence of random variables $X\equiv\{X_i\}_{i\ge 1}$. Let $\mathcal{Q}$ be a family of random probability measures, i.e. each $Q\in\mathcal{Q}$ is a RCP given $X$. Consider the following $\omega$-by-$\omega$ version of \eqref{1}: $$\tag{3}\label{3} \limsup_{n\to \infty}\sup_{Q\in \mathcal{Q}}Q\{A_n\}(\omega)\le \alpha. $$
Under what conditions I may argue that \eqref{3} holds for "almost all" realisations of $X$?