Vol. 41, No. 1, 1972

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A constructive study of measure theory

Yuen-Kwok Chan

Vol. 41 (1972), No. 1, 63–79

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).

Mathematical Subject Classification 2000
Primary: 28A10
Secondary: 02E99
Received: 29 June 1970
Published: 1 April 1972
Yuen-Kwok Chan