On semiregular conditional distributions
Let \((\Omega,{\mathcal A},P)\) denote some probability space and \(\mathcal B\) some sub-\(\sigma\)-algebra of \(\mathcal A\). The existence of a semiregular version of \(P(A|{\mathcal B})\) is studied using a lifting \(L_ 1(\Omega,{\mathcal B},P|{\mathcal B})\to B(\Omega)\). \(B\) is the set of all real bounded functions on \(\Omega\). A strongly semiregular version of \(P(A|{\mathcal B})\) exists if \(L_ 1(\Omega,{\mathcal A},P)\) is separable. The possible decompositions of \(P\) are also formulated.
scientific article; zbMATH DE number 64460