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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $A \\subseteq \\R^n$ be a Lebesgue measurable set with $\\lambda(A) \u003e 0$, where $\\lambda$ is [[Lebesgue measure|2Z2QVWZ262]].\nThen the difference set\n$$A - A = \\set{a - b : a, b \\in A}$$\ncontains an open ball centred at $0$.\n\n## Remarks\n\nOne proof uses inner regularity to find a compact set $K \\subseteq A$ with $0 \u003c \\lambda(K) \u003c \\infty$.\nThe function $f(x) = \\lambda(K \\cap (K + x))$ is continuous and $f(0) = \\lambda(K) \u003e 0$, so $f(x) \u003e 0$ for $x$ near $0$.\nIf $f(x) \u003e 0$, then some $k \\in K$ also lies in $K + x$, so $x = k - k'$ with $k, k' \\in K$, and $x \\in A - A$.\nExample: if $A \\subseteq \\R$ is a measurable set of positive measure, then every real number $t$ with $\\abs{t}$ small enough is a difference of two points of $A$.\nThe hypothesis $\\lambda(A) \u003e 0$ is needed: $A = \\Z$ has measure zero, and $\\Z - \\Z = \\Z$ contains no interval around $0$.\nThe same statement holds in every locally compact group for a measurable set of positive [[Haar measure|VY0HXD3GC4]], with $A A^{-1}$ in place of $A - A$.\n\n## Explanations\n\n### L1 human\n\nTake any set of real numbers that has positive length, however scattered it is.\nThen all small numbers can be written as a difference of two of its points.\nSo a set of positive measure, even with no interval inside it, produces a whole interval of differences.\n\n### L2 human\n\nThe theorem is a key step in automatic continuity results: for instance, a Lebesgue measurable additive function $f \\colon \\R \\to \\R$ is continuous, hence linear.\nIt also shows that a measurable proper subgroup of $\\R$ has measure zero, since a subgroup containing an interval around $0$ is all of $\\R$.\nIts analogue for Baire category is the Pettis lemma.\n",
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