A coin-flip randomized test

The full power set on the one-point data space is a sigma-algebra by [F7], and $P_{{\theta }_0}={\delta }_{x_0}$ is a probability measure by [F1]. Hence the stated family is a statistical experiment by [F2].

The full power set on $\{0,1\}$ is a sigma-algebra by [F7]. The action section assigns masses $0$, $1-p$, $p$, and $1$ to $\mathrm{\varnothing }$, $\{0\}$, $\{1\}$, and $\{0,1\}$, respectively. They are nonnegative and have total mass one by [A1]. For any disjoint sequence of subsets of $\{0,1\}$, each of the two points occurs in at most one member, so the mass of the union equals the sum of the assigned masses. Thus $K(x_0,\cdot )$ is a probability measure.

The data space has one point, so for every action event $B$ the evaluation $x\mapsto K(x,B)$ is constant; its preimages are empty or the whole data space, both measurable. Step 1.2 and this measurability verify that $K$ is a probability kernel and hence a randomized procedure.

By [F4], the test function is $\phi (x_0)=K(x_0,\{1\})=p$. The action-law identity [F5] gives $Q_{{\theta }_0}(\{1\})=\int K(x,\{1\})P_{{\theta }_0}(dx)$. The integrand is the simple function $p{1}_{\mathcal{X}}$, whose integral is $p{P}_{{\theta }_0}(\mathcal{X})=p$ by [F1] and [F6]. This includes both endpoints $p=0$ and $p=1$.