A finitely additive generalization of the Fichtenholz-Lichtenstein theorem

Abstract

Let μ \mu and ν \nu be bounded, finitely additive measures on algebras over sets X and Y, respectively. Conditions are determined for a bounded function f : X × Y → R f:X \times Y \to {\mathbf {R}} , without assuming bimeasurability, so that the iterated integrals ∫ X ∫ Y f d μ d μ \smallint _X {\smallint _Y {fd\mu d\mu } } and ∫ Y ∫ X f d μ d ν \smallint _Y {\smallint _X {fd\mu d\nu } } exist and are equal. This result is then used to construct a product algebra and finitely additive product measure for μ \mu and ν \nu . Finally, a simple Fubini theorem with respect to this product algebra and product measure is established.