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  "body": "\n## Statement\n\nLet $G$ be a [[topological group|92700QAB5Y]] with identity $e$.\nLet $A \\subseteq G$ be a set that has the [[Baire property|FZ7237VY65]] and is not [[meagre|0XGCZ4GDZ6]].\nThen the set $A^{-1} A = \\set{a^{-1} b : a, b \\in A}$ is a [[neighbourhood|0ZCY0Z80B6]] of $e$.\n\n## Remarks\n\nSince $A$ has the Baire property and is not meagre, there is a nonempty open set $U$ such that $A$ differs from $U$ by a meagre set.\nSince $G$ is not meagre, the Banach category theorem and homogeneity show that no nonempty open subset of $G$ is meagre.\nFor $g$ near $e$, the open set $U \\cap gU$ is nonempty, so it is not meagre.\nThe set $A \\cap gA$ differs from it by a meagre set, so $A \\cap gA$ is not empty and $g \\in A A^{-1}$.\nApplying this to $A^{-1}$, which also has the Baire property and is not meagre, gives the statement for $A^{-1}A$.\nIt is the category analogue of the Steinhaus theorem on sets of positive measure.\nA standard application: a homomorphism between Polish groups that is Baire measurable is continuous.\nThe hypothesis that $A$ is not meagre is needed.\nIn $\\R$ the set $\\Z$ is closed, so it has the Baire property, but it is meagre, and $\\Z - \\Z = \\Z$ is not a neighbourhood of $0$.\n\n## Explanations\n\n### L1 human\n\nA set in a group can be large in a topological sense, called not meagre, even when it has no interior.\nThe lemma says that such a set, if it is reasonably well behaved, produces a whole neighbourhood of the identity when you take all quotients $a^{-1} b$ of its elements.\nSo largeness of $A$ turns into an open set inside $A^{-1} A$.\n\n### L2 human\n\nThe lemma is the main tool of automatic continuity results in descriptive set theory.\nTo show a homomorphism $f$ is continuous at $e$, take a neighbourhood $V$ of the identity in the target and a smaller symmetric $W$ with $W^{-1} W \\subseteq V$.\nIf $f^{-1}(W)$ has the Baire property and is not meagre, the lemma shows that $f^{-1}(V)$ contains a neighbourhood of $e$.\n",
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