Decompositions of Substochastic Transition Functions

Three decompositions of a substochastic transition function are shown to yield substochastic parts. These are the Lebesgue decomposition with respect to a finite measure, the decomposition into completely atomic and continuous parts, and on Rn, a decomposition giving a part with continuous distribution function and a part with discontinuous distribution function. Introduction. The present paper considers decomposition theorems for substochastic transition functions. Suppose sé is a cr-algebra of subsets of a set Y. By common usage a substochastic transition function p(y, B) means a function from Yy.se into the unit interval satisfying: (i) for fixed B e sé', y->-p(y, B) is ^-measurable, (ii) for fixed y e Y, p(y, •) is a measure on sé. Now for each y there are several possible decompositions of p(y, ■). If v is a finite measure on sé', then p(y, ■) has a Lebesgue decomposition with respect to v. If sé is countably generated, it makes sense to speak of splitting p(y, •) into completely atomic and continuous parts. Finally, if the underlying space is Euclidean and sé is the collection of Borel sets, it is possible to decompose p(y, •) uniquely into a part having continuous distribution function and a part having discontinuous distribution function, as spelled out in Corollary 4. The point of this paper is to prove that each such decomposition yields substochastic pieces. The combination, in Corollary 9, of these separate decompositions gives a more or less complete breakdown of a substochastic transition function on Euclidean space into four substochastic pieces. Preliminaries and definitions. We wish to modify slightly the definition of substochastic transition function so as to achieve somewhat greater generality and to avoid pathological cases which we are unable to handle. Received by the editors March 3, 1972 and, in revised form, April 20, 1972. AMS (MOS) subject classifications (1970). Primary 60G05, 60J35; Secondary 28A20.