Conditional tests given a statistic

Let $(\mathcal{X},\mathcal{A},\{P_{\theta }{\}}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment with nonempty disjoint null and alternative parameter sets ${\mathrm{\Theta }}_0,{\mathrm{\Theta }}_1\subseteq \mathrm{\Theta }$. Let $\mathcal{S}$ be a topological space with its Borel sigma-algebra $\mathcal{B}(\mathcal{S})$, and let $S:\mathcal{X}\to \mathcal{S}$ be measurable. Write $\sigma (S):=\{S^{-1}(C):C\in \mathcal{B}(\mathcal{S})\}$, the sigma-algebra generated by $S$ as in The sigma-algebra generated by a function. A possibly randomized test is a measurable function $\phi :\mathcal{X}\to [0,1]$ as in Test size and power. For each $\theta \in {\mathrm{\Theta }}_0$, a conditional rejection probability given $S$ is a supplied version $q_{\theta }$ of $E_{\theta }[\phi (X)\mid \sigma (S)]$, where conditional expectation is understood as in Conditional expectation given a sigma algebra. The test has conditional level at most $\alpha$ given $S$ on ${\mathrm{\Theta }}_0$ if such versions can be supplied with $q_{\theta }\le \alpha$ $P_{\theta }$-almost surely for every $\theta \in {\mathrm{\Theta }}_0$. The almost-sure qualification is separate under each null law; no common null set or pointwise statement at conditioning values of probability zero is asserted.