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  "body": "\n## Statement\n\nLet $G$ be a group with a [[standard Borel|ABEFWF4JHH]] structure $\\mathcal{B}$ such that the map $G \\times G \\to G$, $(x, y) \\mapsto x y^{-1}$, is Borel.\nLet $\\mu$ be a nonzero [[σ-finite measure|WEBBT18D4X]] on $\\mathcal{B}$ that is left [[quasi-invariant|FCNQP0A60D]]: for every $g \\in G$ and $A \\in \\mathcal{B}$, $\\mu(gA) = 0$ if and only if $\\mu(A) = 0$.\nThen there is exactly one topology on $G$ with the following two properties.\n\n1. With it, $G$ is a Hausdorff [[locally compact group|AHV05ZBCYW]] that is [[second countable|03PPNCM9GG]].\n2. Its Borel $\\sigma$-algebra is $\\mathcal{B}$.\n\nFor this topology, $\\mu$ has the same null sets as a left [[Haar measure|VY0HXD3GC4]] of $G$.\n\n## Remarks\n\n[[Haar's theorem|KB792QTG4F]] goes from a topology to an invariant measure; this result goes back from an invariant measure to a topology.\nWeil's original version starts from a group with a left-invariant measure that separates points in a suitable sense.\nIt defines a topology from the measure, called the Weil topology.\nThe completion of the group in this topology is locally compact, and the given measure is essentially a Haar measure on it.\nThe form stated here, for standard Borel groups and quasi-invariant measures, is due to Mackey.\nThe idea is that the sets $\\set{g : \\mu(gA \\,\\triangle\\, A) \u003c \\varepsilon}$, for $A$ of finite positive measure and $\\varepsilon \u003e 0$, generate the neighbourhoods of the identity: their finite intersections form a base.\nExample: a Polish group that is not locally compact, such as the additive group of an infinite-dimensional separable Hilbert space, carries no nonzero σ-finite left quasi-invariant Borel measure.\nThis explains why there is no analogue of Lebesgue measure on such a space.\n\n## Explanations\n\n### L1 human\n\nHaar's theorem says that every locally compact group has a way to measure size that does not change under translation.\nThis theorem says the converse.\nIf a reasonable group has a translation-invariant way to measure size, at least up to sets of size zero, then the group is locally compact.\nSo invariant measures and local compactness go together.\n\n### L2 human\n\nThe theorem is used to show that a group has no Haar-like measure: one checks that it cannot be locally compact.\nA standard case is an infinite-dimensional Banach space, which is a Polish group under addition but is not locally compact.\nIn representation theory, the result lets one recover the topology of a locally compact group from its Borel structure and its measure class.\nIt connects to [[quasi-invariant measures|FCNQP0A60D]]: on a second countable locally compact group, every σ-finite left quasi-invariant measure has the same null sets as Haar measure.\n",
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  "msc": [
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