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Functions and measures on product spaces. (English) Zbl 0578.28004

For two completely regular Hausdorff spaces X and Y equipped with Baire measures \(\mu\) and \(\nu\), and for \(f\in C_ b(X\times Y)\), the space of real-valued bounded continuous functions on \(X\times Y\), the paper shows that (i) the map \(Y\to C_ b(X)\) for which \(y\to f(\cdot,y)\) is continuous for the \(\beta\)-topology; (ii) the partial integral \(\int_{x}f(x,y)d\mu\) is continuous for every Y and \(f\in C_ b(X\times Y)\) if and only if \(\mu\) is \(\tau\)-additive; (iii) if \(\mu\) and \(\nu\) are \(\tau\)-additive the Baire sets of \(X\times Y\) are contained in the completion of the standard product \(\sigma\)-algebra with respect to the standard product measure \(\lambda\) ; (iv) if \(\mu\) is \(\sigma\)-additive and \(\nu\) \(\tau\)-additive it is possible to define an extension of \(\lambda\) to the Baire sets of \(X\times Y\) but the Fubini theorem may fail.

MSC:

28A35 Measures and integrals in product spaces
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References:

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