Improper regular conditional distributions (Q1872238)

Let \((\Omega,\mathcal B,P)\) be a probability space and let \(\mathcal A\) be a conditioning sub-\(\sigma\)-field. An rcd (regular conditional distribution) \(P(.|\mathcal A)\) is said to be proper at \(\omega\in {\Omega}\) if \(P(A|{\mathcal A})(\omega)\) = 1 whenever \(\omega\in A\in {\mathcal A}\). If it is not everywhere proper, then it is said to be improper and if \(P(A|{\mathcal A})(\omega)\) = 0 for some \(\omega\in A\in {\mathcal A}\), then it is said to be maximally improper. In this rather technical paper, the authors explore the extent of local and global impropriety. Particular results deal with countably generated resp. symmetric sub-\(\sigma\)-fields. The concluding remarks are interesting.