On an extension of the concept conditional expectation
H. D. Brunk · Proceedings of the American Mathematical Society · 1963
The purpose of the present note is in part to extend results of Sidák [3] to cases where the measure is not finite, perhaps not even <r-finite.In particular, the conditional expectation of a random variable in Lx 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 [l] so as to omit the requirement that the measure be totally finite.For this reason the c-fields discussed in [3] are here replaced by cr-lattices.In this connection it may be observed that E(X\ £) is the solution of the regression problem: given X<ElL2, choose Y in the class C of random variables in L2 measurable with respect to a cr-field £ so as to minimize E(X-Y)2; E(X\ £) 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 £ is not a cr-field but is a cr-lattice, closed under countable union and countable intersection, but not necessarily under complementation (see references in [l]).Let (fi, S, jit) be a measure space: S is a c-field of subsets of fi, and ß is a measure, cr-additive and complete, but not necessarily cr-finite.