Polymeasures are set functions defined on an algebra of subsets of a product space such that the set function is additive, but only componentwise $$\sigma$$-additive. Examples arise in the treatment of certain Feynman integrals. The set function m defined on the family of all products of Borel sets in $${\mathbb{R}}$$ by $m(A\times B)=\int_{A}\int_{B}\phi (x)\exp [i(x-y)^ 2]\psi (y) dx dy$ where $$\phi$$ and $$\psi$$ belong to $$L^ 2({\mathbb{R}})$$ is a bimeasure. It extends uniquely to an additive set function on the algebra generated by all such product sets in $${\mathbb{R}}\times {\mathbb{R}}.$$ However, if either $$\phi$$ or $$\psi$$ does not belong to $$L^ 1({\mathbb{R}})$$, then m is unbounded on this algebra, so it cannot be the restriction of a measure on the generated $$\sigma$$-algebra. - The purpose of this note is to show that there is a useful theory of integration with respect to polymeasures of a certain class. These include those with finitely many components such that the associated (possibly infinite) variation set function is the restriction of a Radon measure on the product space. It is shown that with such an approach, a type of Radon-Nikodým theorem and dominated convergence result are true.