Given two measure spaces, there is always a unique maximal product measure μ_max on their product, with the property that if μ_max(A) is finite for some measurable set A, then μ_max(A) = μ(A) for any product measure μ. In particular its value on any measurable set is at least that of any other product measure. This is the measure produced by the .

Carathéodory extension theorem

Sometimes there is also a unique minimal product measure μ_min, given by μ_min(S) = sup_{A⊂S, μmax(A) finite} μ_max(A), where A and S are assumed to be measurable.

Here is an example where a product has more than one product measure. Take the product X×Y, where X is the unit interval with Lebesgue measure, and Y is the unit interval with counting measure and all sets are measurable. Then, for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either A has Lebesgue measure 0 or B is a single point. (In this case the measure may be finite or infinite.) In particular, the diagonal has measure 0 for the minimal product measure and measure infinity for the maximal product measure.

Fubini's theorem

(1977). "8.2. Product measures and iterated integrals". Probability Theory vol. I (4th ed.). Springer. pp. 135–137. ISBN 0-387-90210-4.

Loève, Michel

(1974). "35. Product measures". Measure theory. Springer. pp. 143–145. ISBN 0-387-90088-8.

Halmos, Paul