  We present a Riesz integral representation theory in which functions,
operators and measures take values in uniform commutative monoids (a
commutative monoid with a uniformity making the binary operation of the monoid
uniformly continuous). It describes the operators to which the theory can be
applied and the finitely-additive measures they generate. Operators satisfying
the conditions will be called ``Riesz integrals''. Given an underlying ``Riesz
system'', it is shown that every Riesz integral generates a certain kind of
finitely additive measure called here a ``Riesz measure''. The correspondence
between Riesz integrals and Riesz measures is a bijection.
  A straightforward calculation shows that if an operator has such a
representation, then it must have the Hammerstein property. For topological
vector spaces, the theory yields necessary and sufficient conditions for
operators with the Hammerstein property to be Riesz integrals. We note that
uniform commutative monoids arise naturally when considering set-valued
functions, and that the axioms of a Riesz system rule out certain spaces of
infinitely differentiable functions.
