The descriptive approach to the derivative of a set function with respect to aσ-lattice

This paper contains a definition and a construction of a Radon-Nikodym derivative of a <r-additive set function with respect to a measure on a ^-lattice, that is, a family of sets closed under countable unions and countable intersections. This derivative is characterized in terms of its indefinite integral, and it is shown how the conditional expectation of an integrable random variable with respect to a <7-lattice, as defined by Brunk, can be obtained as a Radon-Nikodym derivative of the set function determined by the indefinite integral of the random variable.