On some properties of probability kernels

Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the notion of a proper r.c.d. has been introduced. Here, we study how properness, viewed as a property of probability kernels in general, is related to stationarity, compatibility, reversibility and totality, revealing the effects these properties have on the structure of probability kernels. As a further development, we consider the inverse problem of characterizing certain classes of r.c.d.s in terms of the above properties. In particular, we derive necessary and sufficient conditions under which, for a given probability kernel, there exists a unique (in some sense) sub-$\sigma$-algebra such that the probability kernel is a proper r.c.d. given that sub-$\sigma$-algebra.