Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm b\mathscr A$ denote the space of all bounded measurable functions $$ f:(X,\mathscr A)\to(\Bbb R,\mathscr B(\Bbb R)). $$ Let also $\mathcal P(X,\mathscr A)$ denote the collection of all probability measures on $(X,\mathscr A)$.
Definition: we say that collections $P,Q\subseteq \mathcal P(X,\mathscr A)$ are equivalent, and we write $P\sim Q$ if $$ \sup_{p\in P}\int_\Omega f\;\mathrm dp = \sup_{q\in Q}\int_\Omega f\;\mathrm dq \qquad \forall f\in \mathrm b\mathscr A. \tag{1} $$
The relation above is clearly an equivalence relation, and also it does not matter whether we put $\sup$ or $\inf$ in $(1)$. It is useful especially when one wants to show that $Q\subset P$ is a strict subset of $P$, but still $P\sim Q$. However, I am not sure whether that's the only notion of equivalence between the collections of measures, and perhaps $(1)$ already has a different name.