Estimator sequences in IID experiments

Let $\mathsf{P}:\mathrm{\Theta }\to \mathcal{P}(\mathcal{X},\mathcal{A})$ be a parametric model on a statistical experiment, and let $\psi :\mathrm{\Theta }\to \mathcal{T}$ be an estimand whose target set $\mathcal{T}$ is equipped with a metric $d$ (Statistical experiments, Parametric statistical models, Estimands on a statistical model, Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric). For $n\ge 1$, the IID sample-size-$n$ experiment has sample space ${\mathcal{X}}^n$, product sigma-algebra ${\mathcal{A}}^{\otimes n}$, and sampling law $P_{\theta }^{\otimes n}$ at parameter $\theta$ (The product sigma-algebra and its finite iterates). The product law is defined recursively from $P_{\theta }$; each factor is a probability measure and hence sigma-finite, so the finite product theorem supplies its existence, rectangle formula, and uniqueness (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique).

A nonrandomized estimator sequence of $\psi$ is a sequence of measurable maps, in the sense of A measurable function between measurable spaces, $T_n:({\mathcal{X}}^n,{\mathcal{A}}^{\otimes n})\to (\mathcal{T},\mathcal{B}(\mathcal{T})),n\ge 1,$ where $\mathcal{B}(\mathcal{T})$ is the sigma-algebra generated by the open sets of the metric $d$, that is, the Borel sigma-algebra of its metric topology (The Borel sigma-algebra of a topological space). Its sampling distribution at $\theta$ and sample size $n$ is the pushforward of $P_{\theta }^{\otimes n}$ by $T_n$.

This definition fixes IID repeated sampling and data-measurable estimators. It imposes no measurability on the parameter map $\theta \mapsto P_{\theta }$ or on $\psi$.