On product measures and Fubini’s theorem in locally compact space

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1. Introduction. The problems treated in this paper derive from the viewpoint of measure and integration developed in the book of P. R. Halmos [4]. We are concerned, above all, with the formulation of Fubini's theorem in the product of two locally compact spaces, assuming we are given a Borel measure on each of the factor spaces. Our basic tools, treated elegantly in [4], are (1) the theory of the product of two a-finite measure spaces, and (2) the theory of a single Borel measure on a locally compact space. But these tools alone fail to yield a satisfactory Fubini theorem in the context of locally compact spaces. The reason for this failure is that the domain of definition of the product of two Borel measures, as defined in [4], may not be large enough. (Examples are given in ?7 to illustrate insufficient domain. However, a case is given in ?8 in which the domain is sufficient.) To explain this circumstance in greater detail, let us introduce some notations. For the rest of the paper, p and v denote Borel measures on the locally compact spaces X and Y, respectively. (At times we shall assume that ,u or v is regular, or that X = Y.) For the definitions of Borel measure and regular Borel measure, the reader is referred to [4, Chapter X]. Specifically, the Borel sets of X, Y, and X x Y are the a-ring generated by the compact subsets of X, Y, and X x Y, respectively; we denote this class by X(X), X(Y), and X(X x Y), respectively. By the product a-ring of I(X) and X(Y), denoted