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  "body": "\n## Statement\n\nLet $H$ be a real separable [[Hilbert space|HM2HHE3J51]].\nLet $\\mu = N(a, Q)$ and $\\nu = N(b, R)$ be [[Gaussian measures|P0DV2TCH1N]] on $H$ with means $a, b \\in H$ and covariance operators $Q, R$.\nHere $Q$ and $R$ are positive [[self-adjoint|0NEGN5CRSC]] [[trace class|Z2CP530WDE]] operators on $H$.\n\n1. Either $\\mu$ and $\\nu$ are equivalent, that is, each is [[absolutely continuous|P0P21KDV6E]] with respect to the other, or they are [[mutually singular|EBGECZNCV5]].\n2. They are equivalent if and only if the following three conditions hold:\n   (a) $Q^{1/2}(H) = R^{1/2}(H)$; call this space $H_0$.\n   (b) $a - b \\in H_0$.\n   (c) The operator $(Q^{-1/2} R^{1/2})(Q^{-1/2} R^{1/2})^* - I$ is a Hilbert–Schmidt operator on the closure of $H_0$.\n\n## Remarks\n\nHere $Q^{-1/2}$ is the inverse of $Q^{1/2}$ on its image, and $Q^{1/2}(H)$ is the Cameron–Martin space of $\\mu$.\nWhen $Q = R$, the theorem reduces to the Cameron–Martin theorem: $N(a, Q)$ and $N(b, Q)$ are equivalent exactly when $a - b$ lies in the Cameron–Martin space.\nThe proof compares the two measures along an orthonormal basis and applies Kakutani's theorem on infinite products of measures, which also has an all-or-nothing form.\n\nExample: on $\\R^n$, two Gaussian measures with invertible covariance matrices are always equivalent, since both have everywhere positive densities.\nCondition (c) is automatic in finite dimensions.\n\nExample: on an infinite-dimensional $H$ with $Q$ injective, $N(0, Q)$ and $N(0, c Q)$ with $c \u003e 0$, $c \\ne 1$, are mutually singular.\nIndeed the operator in (c) is $(c - 1) I$, which is not Hilbert–Schmidt.\n\n## Explanations\n\n### L1 human\n\nIn finite dimensions, two non-degenerate Gaussian laws have the same null sets.\nIn infinite dimensions this is false, and two Gaussian measures either have exactly the same null sets or live on disjoint sets.\nThe theorem says precisely when the first case happens.\n\n### L2 human\n\nThe dichotomy is a basic fact for statistics of Gaussian processes.\nIf two models give singular laws, one can tell them apart without error from a single infinite-dimensional observation.\nFor example, the variance parameter of Brownian motion can be recovered exactly from one path, which matches the scaling example above.\n",
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