Product measure

Summary

The product measure $\mu \times \nu$ of two σ-finite measures is the measure on the product σ-algebra that gives each rectangle $A \times B$ the measure $\mu(A) \nu(B)$. It exists and is unique under σ-finiteness. It is the setting of Fubini's and Tonelli's theorems, which compute integrals over the product as iterated integrals. In probability it describes pairs of independent random variables.