A note on a representation of a transition by pivotal measure (Q1073471)

If \({\mathcal E}=(\Omega,{\mathcal A},{\mathcal P})\) denotes a weakly dominated statistical experiment (i.e. \({\mathcal P}\) is absolutely continuous with respect to a localizable measure) a measure n equivalent to \({\mathcal P}\) is pivotal if and only if for each sufficient sub-\(\sigma\)-field \({\mathcal B}\) of \({\mathcal A}\) every \(P\in {\mathcal P}\) has a \({\mathcal B}\)-measurable density w.r.t.\ n. It is shown that the property of n to be pivotal is equivalent to \[ n(A)=\int E(I_ A| {\mathcal B})d(n| {\mathcal B})\quad for\quad all\quad A\in {\mathcal A}. \] This result is applied to find a transition from \({\mathcal E}({\mathcal B})=(\Omega,{\mathcal B},{\mathcal P}| {\mathcal B})\) to \({\mathcal E}\) (i.e. a positive, linear and positively isometric mapping) such that the deficiency of \({\mathcal E}({\mathcal B})\) w.r.t.\ \({\mathcal E}\), which is defined by \(\inf \{\sup (\| T(P| {\mathcal B})-P\|;\quad P\in {\mathcal P});\quad T:{\mathcal E}({\mathcal B})\to {\mathcal E}\) transition\(\}\), is zero.