In this paper we study
measures on locally compact metric spaces. The constructive theory of a
nonnegative measure has been treated in Bishop’s book “Foundations of
Constructive Analysis”. Unfortunately, there is no constructive method to
decompose a general signed measure into a difference of two nonnegative ones. In
analogy to the classical development, we shall consider two ways to look
at a signed measure, namely, as a function function (an integral) and as a
set function (a set measure). From an integral on a locally compact metric
space X we obtain compact subsets of X to which measures can be assigned.
The set measure thus arrived at is shown to be in a weak sense additive,
continuous, and of bounded variation. Next we study a set measure having
these three properties defined on a large class of compact subsets of X.
From such a set measure we derive a linear function on the space of test
functions of X. This linear function is then shown to be an integral. Finally it is
demonstrated that the set measure arising from an integral gives rise in this
manner to an integral which is equal to the original one. In particular, every
integral is the integral arising from some measure (Riesz Representation
Theorem).
