  We show that, if S is a finite semiring, then the free profinite S-semimodule
on a Boolean Stone space X is isomorphic to the algebra of all S-valued
measures on X, which are finitely additive maps from the Boolean algebra of
clopens of X to S. These algebras naturally appear in the logic approach to
formal languages as well as in idempotent analysis. Whenever S is a (pro)finite
idempotent semiring, the S-valued measures are all given uniquely by continuous
density functions. This generalises the classical representation of the
Vietoris hyperspace of a Boolean Stone space in terms of continuous functions
into the Sierpinski space.
  We adopt a categorical approach to profinite algebra which is based on
profinite monads. The latter were first introduced by Adamek et al. as a
special case of the notion of codensity monads.
