An inequality for the Hausdorff-metric of \(\sigma\)-fields
The authors prove the following inequality, and then provide a number of consequent corollaries: Theorem: Let \({\mathcal A},{\mathcal B}\subset {\mathcal F}\) be \(\sigma\)-fields and let \(A_ i\in {\mathcal A}\), \(i\in I\), be disjoint sets. Then \[ \sum_{i\in I}{\tilde \rho}(A_ i,{\mathcal B})\leq 4\rho ({\mathcal A},{\mathcal B}) \] where \(\rho\) is the pseudometric on sub \(\sigma\)-fields of a given probability space, (\(\Omega\),\({\mathcal F},P)\), defined by: a) \({\tilde \rho}\)(A,\({\mathcal B})=\inf_{B\in {\mathcal B}}P(A\Delta B)\), (\(\Delta\) represents the symmetric difference operator); b) \({\tilde \rho}\)(\({\mathcal A},{\mathcal B})=\sup_{A\in {\mathcal A}}{\tilde \rho}(A,{\mathcal B})\); c) \(\rho\) (\({\mathcal A},{\mathcal B})={\tilde \rho}({\mathcal A},{\mathcal B})+{\tilde \rho}({\mathcal B},{\mathcal A})\).
scientific article; zbMATH DE number 3962839