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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ and $(Y, T, \\nu)$ be measure spaces, and assume that $\\mu$ and $\\nu$ are [[σ-finite|WEBBT18D4X]].\nLet $\\Sigma \\otimes T$ be the [[product σ-algebra|6HVCTJGB61]] on $X \\times Y$.\nThe *product measure* $\\mu \\times \\nu$ is the unique [[measure|PR11BZQQPG]] on $\\Sigma \\otimes T$ with\n$$(\\mu \\times \\nu)(A \\times B) = \\mu(A) \\, \\nu(B) \\quad \\text{for all } A \\in \\Sigma, \\ B \\in T,$$\nwith the convention $0 \\cdot \\infty = 0$.\n\n## Remarks\n\nExistence follows from [[Carathéodory's extension theorem|Q8EXG7EKYW]] applied to the rectangles, and uniqueness from σ-finiteness and the π-λ theorem.\nFor $E \\in \\Sigma \\otimes T$ one has $(\\mu \\times \\nu)(E) = \\int_X \\nu(E_x) \\, d\\mu(x) = \\int_Y \\mu(E^y) \\, d\\nu(y)$, where $E_x$ and $E^y$ are the sections of $E$.\n\nExample: let $\\mu$ and $\\nu$ be Lebesgue measure on the Borel sets of $[0, 1]$.\nThen $\\mu \\times \\nu$ is two-dimensional Lebesgue measure on the Borel sets of $[0, 1]^2$: both measures give each rectangle its area, so they agree by uniqueness.\n\nNon-example: let $\\delta$ be the image of Lebesgue measure $\\lambda$ on $[0, 1]$ under $x \\mapsto (x, x)$, a measure on the diagonal of $[0, 1]^2$.\nFor $A = [0, \\tfrac12]$ and $B = [\\tfrac12, 1]$, $\\delta(A \\times B) = \\lambda(A \\cap B) = 0$, but $\\lambda(A) \\lambda(B) = \\tfrac14$.\nSo $\\delta$ is not $\\lambda \\times \\lambda$.\n\nWithout σ-finiteness, uniqueness can fail.\nFor Lebesgue measure on $[0, 1]$ and counting measure on $[0, 1]$, there are different measures on the product σ-algebra with the rectangle property.\nThe product of Lebesgue measures is often completed to give Lebesgue measure on $\\R^{m+n}$.\n",
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