A generalized Radon-Nikodym derivative

Abstract

Let {v α , ae R] be a family of signed measures on a cr-field S/ of subsets of an abstract space Ω.Let ^ be a sub ΰlattice of S/\ Under certain conditions we associate with the family of measures and ^ί a function /, which we call the Lebesgue-Radon-Nikodym (LRN) function.The function / is measurable ^# and satisfies the relations v a (Bn [f< a]) ^ 0 , aeR, [/>&]) ^0, beR, This paper contains a construction of / by means of a Jordan-Hahn decomposition for <7-lattices, and gives various characterizations and representations of /.Special cases are: the derivative of a signed measure with respect to a nonnegative measure, conditional expectation given a <7-field, and conditional expectation given a ^-lattice.The LRN function also provides a conditional generalized mean whose relationship to the generalized mean parallels the relationship of the conditional expectation to the expectation.The paper also contains a convergence theorem for LRN functions with respect to an increasing sequence of tf-lattices, thus generalizing the martingale convergence theorem.Finally it is proved that / is the solution to a minimization problem, generalizing known minimizing properties of conditional expectation and of conditional expectation given a (T-lattice.These properties exhibit the latter as solution of various problems of restricted maximum likelihood estimation.