On an extension of the concept conditional expectation

Abstract

The purpose of the present note is in part to extend results of Sidak [3] to cases where the measure is not finite, perhaps not even r-finite. In particular, the conditional expectation of a random variable in L1 is obtained without the use of the Radon-Nikodym Theorem, which, indeed, does not apply with the required generality. It was found possible to extend at the same time results of [1 ] so as to omit the requirement that the measure be totally finite. For this reason the or-fields discussed in [3] are here replaced by c-lattices. In this connection it may be observed that E(XI ?) is the solution of the regression problem: given XCL2, choose Y in the class C of random variables in L2 measurable with respect to a c-field 4 so as to minimize E(XY)2; E(XI 4) is the projection in L2 of X on C. Certain problems of maximum likelihood estimation of ordered parameters have solutions which also solve the above regression problem, in which 4 is not a c-field but is a c-lattice, closed under countable union and countable intersection, but not necessarily under complementation (see references in [1 ]). Let (Q, 8, ,) be a measure space: S is a c-field of subsets of Q, and ,u is a measure, c-additive and complete, but not necessarily cr-finite. The symbol S will also be used to denote the class of all events: an event is an equivalence class of sets in 8, two sets being equivalent if their symmetric difference has measure 0. The symbol Q will denote also the equivalence class to which Q belongs. (Events will often be denoted by descriptions enclosed in brackets.) A random variable is an equivalence class of real-valued S-measurable functions on Q, two such functions representing the same random variable if they differ on a set of measure 0. If A is an event, Ac will denote the complement of A, and x(A) or XA will denote the indicator random variable of A. Let L1 denote the class of integrable random variables, L2 the class of square-integrable random variables. If 4 is a sub-cr-lattice (closed under countable union, countable intersection) of S containing 0 and Q, let a random variable X be termed ?-measurable if [X > a] E 4 for every real a. A family C of random variables will be called a convex cone if k_0, XCC, YCEC=kXC-C, X+YCC; a