Randomized conditional upper-tail p-values

Fix one null probability law $P_0$. Let $S$ be a random element of a measurable space and let $T$ be a real random variable on the same probability space. Suppose a conditional law kernel $K(s,\cdot )$ of $T$ given $S$ is supplied, in the sense of Conditional law given a random element. On an auxiliary extension, let $U$ be independent of $(S,T)$ and have the uniform Borel probability law on $[0,1]$, characterized by $P(U\le u)=u$ for $0\le u\le 1$. The randomized conditional upper-tail p-value is $p_U:=K(S,(T,\mathrm{\infty }))+UK(S,\{T\}),$ and its nonrandomized upper-tail value is $p_0:=K(S,[T,\mathrm{\infty }))\mathrm{.}$ The strict tail and atom terms are evaluated at the realized pair $(S,T)$; the independent $U$ randomizes only within the tie mass. Their measurability, including measurability of both displayed p-values, follows from Conditional-kernel tail and atom evaluations are measurable. The inclusive tail is the sum of the strict tail and the atom at $T$. This definition asserts no existence of the conditional kernel; it is supplied as part of the input.