Vol. 42, No. 3, 1972

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On Borel product measures

Woodrow Wilson Bledsoe and Charles Edward Wilks

Vol. 42 (1972), No. 3, 569–579

It has been known for many years that the product of two regular borel measures on compact hausdorff topological spaces may not be borel in the product topology. The problem of defining a new product measure that extends the classical product measure and carries over this borel property has been approached in different ways by Edwards, by Bledsoe and Morse (Product Measures, Trans. Amer. Math. Soc. 79 (1955), 173215; called PM here.) and by Johnson and Berberian. Godfrey and Sion and Hall have shown that all three of these methods are equivalent for the case of Radon measures on locally compact hausdorff spaces.

Elliott has extended the results of PM by defining a product measure for a pair, the first of which is a (generalized) borel measure and the second a continuous regular conditional measure (generalization of conditional probability), and proving a corresponding Fubini-type theorem.

The purpose of this paper is to extend the results of PM in a manner similar to Elliott’s, but with his continuity condition replaced by an absolute continuity condition and by a “separation of variables” condition. It is still an open question whether Elliott’s continuity condition is necessary.

Mathematical Subject Classification 2000
Primary: 28A35
Received: 20 May 1971
Published: 1 September 1972
Woodrow Wilson Bledsoe
Charles Edward Wilks