##
**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 |

### Keywords:

measures on product spaces; completely regular Hausdorff spaces; Baire measures; product measure
PDFBibTeX
XMLCite

\textit{A. G. Babiker} and \textit{J. D. Knowles}, Mathematika 32, 60--67 (1985; Zbl 0578.28004)

Full Text:
DOI

### References:

[1] | Bourbaki, Integration (1969) |

[2] | Bourbaki, General Topology (1966) |

[3] | DOI: 10.2307/1992844 · Zbl 0064.29103 |

[4] | Varadarajan, Mat. Sb., NS 55 pp 33– (1961) |

[5] | DOI: 10.2307/1996178 · Zbl 0244.46027 |

[6] | DOI: 10.1007/BF02591355 · Zbl 0147.04501 |

[7] | Comfort, Math. Scand 28 pp 77– (1971) · Zbl 0217.47904 |

[8] | DOI: 10.1112/plms/s3-17.1.139 · Zbl 0154.05101 |

[9] | DOI: 10.1007/BF01896008 · Zbl 0234.04009 |

[10] | DOI: 10.2307/2037558 · Zbl 0229.28006 |

[11] | Gillman, Rings of continuous functions (1960) |

[12] | DOI: 10.1112/plms/s3-25.1.115 · Zbl 0236.46025 |

[13] | Fremlin, Canad. Math. Bull 19 pp 285– (1976) · Zbl 0353.28005 |

[14] | Ressel, Math. Scand 40 pp 69– (1977) · Zbl 0372.28010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.