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  "body": "\n## Statement\n\nLet $T$ be a nonempty index set, and for each $t \\in T$ let $E_t$ be a [[Polish space|584GXXY73M]] with its Borel σ-algebra.\nFor $J \\subseteq T$ let $E_J = \\prod_{t \\in J} E_t$ with the [[product σ-algebra|6HVCTJGB61]], and for $K \\subseteq J$ let $\\pi^J_K \\colon E_J \\to E_K$ be the restriction of coordinates.\nFor each finite nonempty $J \\subseteq T$ let $\\mu_J$ be a [[probability measure|059TJN0NZ1]] on $E_J$.\nAssume the family is consistent: for all finite nonempty $K \\subseteq J \\subseteq T$, the [[pushforward|3N66TM3TSQ]] of $\\mu_J$ under $\\pi^J_K$ equals $\\mu_K$.\nThen there is a unique probability measure $\\mu$ on $E_T$ such that, for every finite nonempty $J \\subseteq T$, the pushforward of $\\mu$ under $\\pi^T_J$ equals $\\mu_J$.\n\n## Remarks\n\nThe measure $\\mu$ is first defined on cylinder sets, where consistency makes it well defined and finitely additive.\nThe key step is countable additivity on the algebra of cylinder sets.\nIt uses inner regularity: each $\\mu_J$ is approximated from inside by compact sets, and a decreasing sequence of nonempty compact cylinders has nonempty intersection.\n[[Carathéodory's extension theorem|Q8EXG7EKYW]] then extends $\\mu$ to the product σ-algebra, and uniqueness holds because cylinder sets form a π-system.\n\nExample: let $T = [0, \\infty)$, $E_t = \\R$, and let $\\mu_J$ be the centred Gaussian measure on $\\R^J$ with covariance $\\min(s, t)$ for $s, t \\in J$.\nThese measures are consistent, so there is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \\ge 0}$ with these finite-dimensional laws.\nThis is the first step in the construction of Brownian motion; the continuity of paths needs a further argument.\n\nA topological assumption on the factors is needed.\nFor arbitrary measurable spaces in place of Polish spaces, consistent families exist that have no extension.\nNo such assumption is needed when the $\\mu_J$ are products, as for the [[infinite product measure|2HWVFHM13R]].\nIt is not needed either when they are built from kernels, as in the [[Ionescu-Tulcea theorem|94ASD187AQ]].\n\n## Explanations\n\n### L1 human\n\nTo describe a random process, such as the position of a particle at every time, it is enough to say how it behaves at finitely many times at once.\nIf these finite descriptions do not contradict each other, the theorem gives a single probability measure on the whole space of paths that matches all of them.\nSo a process can be built from its finite-dimensional distributions.\n\n### L2 human\n\nThe theorem is the standard existence result for stochastic processes with prescribed finite-dimensional distributions: Gaussian processes, Markov processes from transition functions, and sequences of independent variables.\nThe resulting σ-algebra only contains events that depend on countably many coordinates.\nSo for uncountable $T$, sets such as \"the path is continuous\" are not measurable, and one has to pass to a modification, for instance with the [[Kolmogorov continuity theorem|V1HAJNC5RK]].\n",
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