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  "body": "\n## Statement\n\nLet $X$ be an infinite-dimensional separable real [[Banach space|06GZ956657]].\nLet $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$ such that these conditions hold:\n\n1. $\\mu$ is translation-invariant: $\\mu(x + A) = \\mu(A)$ for every Borel set $A$ and every $x \\in X$.\n2. $\\mu$ is locally finite: every point of $X$ has an open neighbourhood of finite measure.\n\nThen $\\mu(A) = 0$ for every Borel set $A$.\n\n## Remarks\n\nBy clause 1 and clause 2, some [[open ball|0DC3MPR7C9]] $B(0, r)$ with $r \u003e 0$ has finite measure.\nBy [[Riesz's lemma|097C7KCAT5]], there are infinitely many points $x_1, x_2, \\ldots$ with $\\norm{x_i} = 3r/4$ and $\\norm{x_i - x_j} \\ge r/2$ for $i \\ne j$.\nThe balls $B(x_i, r/4)$ are disjoint, lie in $B(0, r)$, and all have the same measure by translation invariance; so this measure is $0$.\nBy separability, $X$ is a countable union of balls of radius $r/4$, and so $\\mu(X) = 0$.\n\nThe result is often stated in an equivalent way: a translation-invariant Borel measure that is not identically zero gives infinite measure to every nonempty open set.\nIn finite dimension the conclusion fails, because [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$ is translation-invariant and locally finite.\nGaussian measures on infinite-dimensional spaces avoid the theorem by giving up translation invariance.\n\n## Explanations\n\n### L1 human\n\nIn $\\R^n$ volume does not change when a set is moved, and small balls have finite volume.\nIn infinite dimension no notion of volume can have both properties, except the one that gives volume zero to everything.\nThe reason is that a ball contains infinitely many disjoint smaller balls of a fixed size, which would all need the same positive volume.\n\n### L2 human\n\nThe theorem explains why integration over spaces of functions, as in probability theory and quantum field theory, uses other measures.\nThe standard choices are Gaussian measures, such as Wiener measure on paths.\nTheir behaviour under translations is described by the Cameron–Martin theorem: only translations along a small subspace keep the measure class.\n",
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