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  "body": "\n## Definition\n\nLet $I$ be a nonempty index set and, for each $i \\in I$, let $(X_i, \\Sigma_i, \\mu_i)$ be a [[probability space|0NFPJVJJHX]].\nLet $X = \\prod_{i \\in I} X_i$ with the [[product σ-algebra|6HVCTJGB61]] $\\Sigma = \\bigotimes_{i \\in I} \\Sigma_i$.\nThe *product measure* $\\bigotimes_{i \\in I} \\mu_i$ is the unique [[probability measure|059TJN0NZ1]] $\\mu$ on $(X, \\Sigma)$ with\n$$\\mu\\Big( \\prod_{i \\in J} A_i \\times \\prod_{i \\in I \\setminus J} X_i \\Big) = \\prod_{i \\in J} \\mu_i(A_i)$$\nfor every finite $J \\subseteq I$ and all $A_i \\in \\Sigma_i$, $i \\in J$.\n\n## Remarks\n\nSuch a measure exists for every family of probability spaces, without topological assumptions, and it is unique because the cylinder sets form a π-system that generates $\\Sigma$.\nFor countable $I$, existence also follows from the [[Ionescu-Tulcea theorem|94ASD187AQ]] with kernels that do not depend on the past.\n\nExample: let $X_n = \\set{0, 1}$ and $\\mu_n(\\set{0}) = \\mu_n(\\set{1}) = \\tfrac12$ for $n = 1, 2, 3, \\ldots$\nThe product measure on the space of sequences $(x_1, x_2, \\ldots)$ with $x_n \\in \\set{0, 1}$ is the law of a sequence of fair coin tosses.\nThe set of sequences starting with $0, 1$ is a cylinder set with $J = \\set{1, 2}$ and measure $\\tfrac12 \\cdot \\tfrac12 = \\tfrac14$.\n\nNon-example: on the same space, the Dirac measure at the constant sequence $(0, 0, 0, \\ldots)$ is a probability measure.\nIt gives the cylinder set $\\set{x : x_1 = 0}$ measure $1$ instead of $\\tfrac12$, so it is not the product of the fair coin measures.\n\nUnder the product measure, the coordinate maps $\\pi_i$ are independent random variables, and $\\pi_i$ has law $\\mu_i$.\n",
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