A finitely additive generalization of the Fichtenholz-Lichtenstein theorem

Abstract:

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