Products of two Borel measures

Johnson, Roy A.

doi:10.1090/S0002-9947-1982-0637713-6

Products of two Borel measures
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Trans. Amer. Math. Soc. 269 (1982), 611-625 Request permission

Abstract:

Let $\mu$ and $\nu$ be finite Borel measures on Hausdorff spaces $X$ and $Y$, respectively, and suppose product measures $\mu \times {}_1\nu$ and $\mu \times {}_2\nu$ are defined on the Borel sets of $X \times Y$ by integrating vertical and horizontal cross-section measure, respectively. Sufficient conditions are given so that $\mu \times {}_1\nu = \mu \times {}_2\nu$ and so that the usual product measure $\mu \times \nu$ can be extended to a Borel measure on $X \times Y$ by means of completion. Examples are given to illustrate these ideas.

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