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| In the mathematical field of [[real analysis]], the '''Steinhaus theorem''' states that the [[difference set]] of a set of positive [[measure (mathematics)|measure]] contains an [[open set|open]] [[neighbourhood (mathematics)|neighbourhood]] of zero. It was first proved by [[Hugo Steinhaus]].<ref>{{harvtxt|Steinhaus|1920}}; {{harvtxt|Väth|2002}}</ref>
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| ==Statement==
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| Let ''A'' be a Lebesgue-measurable set on the [[real line]] such that the [[Lebesgue measure]] of ''A'' is not zero. Then the ''difference set''
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| : <math>A-A=\{a-b\mid a,b\in A\} \, </math>
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| contains an open neighbourhood of the origin.
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| More generally, if ''G'' is a [[locally compact group]], and ''A'' ⊂ ''G'' is a subset of positive (left) [[Haar measure]], then
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| : <math> AA^{-1} = \{ ab^{-1} \mid a,b \in A \} \, </math>
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| contains an open neighbourhood of unity.
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| The theorem can also be extended to [[meagre set|nonmeagre]] sets with the [[Baire property]]. The proof of these extensions, sometimes also called Steinhaus theorem, is almost identical to the one below.
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| ==Proof==
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| The following is a simple proof due to Karl Stromberg.<ref>{{harvtxt|Stromberg|1972}}</ref>
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| If ''μ'' is the Lebesgue measure and ''A'' is a measurable set with positive finite measure
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| :<math>0<\mu(A)<\infty,\,</math>
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| then for every ''ε'' > 0 there are a [[compact set]] ''K'' and an open set ''U'' such that
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| :<math> K\subset A \subset U, \quad \mu (K)+\epsilon>\mu(A)>\mu(U)-\epsilon.</math>
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| For our purpose it is enough to choose ''K'' and ''U'' such that
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| :<math>2\mu (K)>\mu(U).\,</math>
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| Since ''K'' ⊂ ''U'', there is an [[open cover]] of ''K'' that is contained in ''U''. ''K'' is compact, hence one can choose a small neighborhood ''V'' of 0 such that ''K'' + ''V'' ⊂ ''U''.
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| Let ''v'' ∈ ''V'', and suppose
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| :<math> (K+v)\cap K=\varnothing.\,</math> | |
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| Then,
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| :<math> 2\mu(K)=\mu(K+v)+\mu(K)<\mu(U)\,</math>
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| contradicting our choice of ''K'' and ''U''. Hence for all ''v'' ∈ ''V'' there exist
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| : <math>k_{1}, k_{2}\in K \subset A\,</math>
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| such that
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| :<math>v+k_{1}=k_{2},\,</math>
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| which means that ''V'' ⊂ ''A'' − ''A''. [[Q.E.D.]]
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| ==See also==
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| *[[Falconer's conjecture]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{citation
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| | last = Steinhaus | first = Hugo | author-link = Hugo Steinhaus
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| | journal = Fund. Math.
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| | language = French
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| | pages = 93–104
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| | title = Sur les distances des points dans les ensembles de mesure positive
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| | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1111.pdf
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| | volume = 1
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| | year = 1920}}.
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| * {{cite journal|last=Stromberg|first=K.|jstor=2039082|title=An Elementary Proof of Steinhaus's Theorem|journal=Proceedings of the American Mathematical Society|volume=36|issue=1|year=1972|page=308|ref=harv}}
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| * {{cite book
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| | last = Väth
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| | first = Martin,
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| | title = Integration theory: a second course
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| | publisher = World Scientific
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| | date = 2002
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| | pages =
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| | isbn = 981-238-115-5
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| }}
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| [[Category:Theorems in measure theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in real analysis]]
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