Product measure - Wikipedia

Abstraction: Measure on Cartesian product of measurable spaces extending component measures

Key points:

• Given two measurable spaces (X, A, mu) and (Y, B, nu), the product sigma-algebra is generated by rectangles A x B; a product measure satisfies (mu x nu)(A x B) = mu(A) * nu(B)
• Product measure is unique when both spaces are sigma-finite (guaranteed by the Hahn-Kolmogorov theorem); non-sigma-finite spaces can have multiple product measures
• Borel measures on R^n are the product of n copies of the Borel measure on R; extending to Lebesgue measure requires a completion step
• There is always a unique maximal product measure (from Caratheodory extension) and sometimes a unique minimal product measure; non-sigma-finite spaces can have a continuum of product measures between them
• Disintegration is the opposite construction: it splits a measure into a family of measures that integrate back to the original

Connections: Measure Theory · Product Measure · Sigma Algebra

Source: http://en.wikipedia.org/wiki/Product_measure