Pivotal quantities

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment with $\mathrm{\Theta }\mathrm{\ne }\mathrm{\varnothing }$, and let $(S,\mathcal{S})$ be a measurable space. A family of measurable maps $q_{\theta }:\mathcal{X}\to S$, one for each $\theta \in \mathrm{\Theta }$, is a pivot if the pushforward law ${\nu }_{\theta }(B):=P_{\theta }(q_{\theta }^{-1}(B)),B\in \mathcal{S}$ is the same probability measure $\nu$ for every $\theta \in \mathrm{\Theta }$. Each section is a random element on $(\mathcal{X},\mathcal{A},P_{\theta })$, so its pushforward is a probability measure by The law of a random element is a probability measure. No measurability in $\theta$ is asserted or needed.

For $S=\mathbb{R}$, a pivot is a parameter-indexed family of measurable real-valued quantities with a common sampling law. The maps $q_{\theta }$ may depend on the unknown parameter. A pivot need not be invertible in the parameter, and its inversion need not be an interval.