Product Measure

Let (X,A,mu) and (Y,B,nu) be measure spaces, let R be the collection of all measurable rectangles contained in X×Y, and let lambda be the premeasure defined on R by


for A×B in R. By the product measure lambda=mu×nu, one means the Carathéodory extension of lambda:R->[0,infty] defined on the sigma-algebra of (mu×nu)^*-measurable subsets of X×Y where (mu×nu)^* denotes the outer measure induced by the premeasure mu×nu on R.

See also

Carathéodory Extension, Carathéodory Measure, Measurable Function, Measurable Rectangle, Measurable Set, Measurable Space, Measure, Measure Space, Outer Measure, Premeasure

This entry contributed by Christopher Stover

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Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.

Cite this as:

Stover, Christopher. "Product Measure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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