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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), 173−215; 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.
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