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  "body": "\n## Summary\n\nThe product measure $\\mu \\times \\nu$ of two [[σ-finite|WEBBT18D4X]] measures is the measure on the [[product σ-algebra|6HVCTJGB61]] that gives each rectangle $A \\times B$ the measure $\\mu(A) \\nu(B)$.\nIt exists and is unique under σ-finiteness.\nIt is the setting of Fubini's and Tonelli's theorems, which compute integrals over the product as iterated integrals.\nIn probability it describes pairs of independent random variables.\n\n## Explanations\n\n### L1 human\n\nThe area of a rectangle is width times height.\nThe product measure extends this rule from rectangles to all reasonable subsets of a product space.\nLebesgue measure on the plane is the product of Lebesgue measure on the line with itself.\nIf one throws a fair coin and a fair die independently, the probability of \"heads and a six\" is $\\tfrac12 \\cdot \\tfrac16 = \\tfrac{1}{12}$, a value of the product measure.\n",
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