An open action set need not contain a posterior-risk minimizer

The prior and data law are point masses on singleton standard-Borel spaces by [F1]; their joint law is the point mass on the singleton pair. Define the posterior to be the point mass at $0$. The only conditioning events are empty or the whole singleton, and the conditional-law identity holds for both, so this is a regular conditional law by [F3]. Integrating $L(0,a)=a$ against it gives posterior risk $q(a)=a$ by [F2].

For every $a\in (0,1)$, $q(a)=a>0$, while the sequence $a_n=1\mathrm{/}(n+1)$ lies in $(0,1)$ and has $q(a_n)\to 0$. Thus ${\inf }_{a\in (0,1)}q(a)=0$ and the infimum is not attained. The claimed universal minimizer does not exist, which refutes the statement and illustrates the explicit existence premise in [F4].