Disintegration with respect to L p-density functions and singular measures

Summary Let A be a real or complex commutative ordered algebra with identity and involution. Let denote the set of positive multiplicative linear functionals on A. Equip with the topology of simple convergence. For a fixed non-negative probability measure on the set _p of linear functionals f on A which admit an integral representation of the form with FL_p() (1p) is biuniquely identified with L_p() via the map tfF. The norm on _p under which this map becomes an isometry is characterized and a formula for approximating F is derived. The linear functionals which admit representation of the form with are also characterized and appropriately normed. The theory is applied to solve abstract versions of trigonometric and n-dimensional moment problems as well as provide an alternate point of view to the theory of L_p-spaces. New proofs of classical theorems are offered.Research for this paper was sponsored in part by the Danish Natural Science Research Council (Grant No.511-10302) and in part by the National Science Foundation (Grant No. MCS78-03397)The results contained herein include the proofs of theorems announced in [15]