Given a Polish space $E$ endowed with Borel $\sigma$-algebra. We consider a sequence of empirical measures $(\eta_N)_{N> 0}$ such that for all the bounded measurable test functions $f\in\mathcal{B}_b(E)$, we have $$ \eta_N (f) \xrightarrow[N\rightarrow \infty]{a.s.}\eta(f) $$ Now, let us consider the $\eta\textrm{-ess sup}(f)$ defined by $$ \eta\textrm{-ess sup}(f) := \inf\{a \in \mathbb{R}: \eta (\{x:f(x)>a\}=0)\} $$ Fixing some $f\in \mathcal{B}_b(E)$, the essential supremum associated to the empirical measures could then be seen as a sequence of random variables. More precisely, let us define $$ X_N :=\eta_N\textrm{-ess sup}(f) \qquad N\in \mathbb{N}^* $$ My question: Can we have some convergence results for the sequence of random variables $(X_N)_{N>0}$. For example: $$ X_N \xrightarrow[N\rightarrow \infty]{\mathbb{P} \textrm{ or } a.s.}\eta\textrm{-ess sup}(f) $$ If not, what may possibly be the missing assumption?
Thanks in advance!