Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces. The existence of this measure is guaranteed by the Hahn–Kolmogorov theorem. The uniqueness of product measure is guaranteed only in case that both (X1, 1, 1) and (X2, 2, 2) are -finite. The Borel measure on the Euclidean space Rn can be obtained as the product of n copies of the Borel measure on the real line R. Even if the two factors of the product space are complete measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure, or to extend the product of two Lebesgue measures to give the Lebesgue measure on the product space.

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