Randomized tests

A randomized test is a randomized procedure with binary action space whose action set is $\{0,1\}$, equipped with its full sigma-algebra $2^{\{0,1\}}$ (the power set, which is a sigma-algebra by Sigma-algebras), where $1$ means reject and $0$ means do not reject (Randomized procedures as probability kernels). Its test function is $\phi (x)=K(x,\{1\}),x\in \mathcal{X}\mathrm{.}$ Since $K(x,\cdot )$ is a probability measure and $\{0\},\{1\}$ partition the action set, $K(x,\{0\})+K(x,\{1\})=1$. Thus $\phi (x)\in [0,1]$ and $1-\phi (x)=K(x,\{0\})$ is the probability of not rejecting. Give $[0,1]$ the trace of the Borel sigma-algebra on $[0,+\mathrm{\infty }]$, namely $\{C\cap [0,1]:C\in \mathcal{B}([0,+\mathrm{\infty }])\}$. By A randomized procedure has a measurable action law, $x\mapsto K(x,\{1\})$ is measurable; since it has range in $[0,1]$, its preimage of $C\cap [0,1]$ equals its preimage of $C$ and is measurable for every such $C$. Hence $\phi$ is a measurable map in the preimage sense of A measurable function between measurable spaces. Its value is the rejection probability after observing $x$. The convention $1=\text{reject}$ is fixed here. No null or alternative hypothesis is part of this definition.