# Steinhaus theorem

Steinhaus theorem In the mathematical field of real analysis, the Steinhaus theorem states that the difference set of a set of positive measure contains an open neighbourhood of zero. It was first proved by Hugo Steinhaus.[1] Contents 1 Statement 2 Proof 3 Corollary 4 See also 5 Notes 6 References Statement 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 {displaystyle A-A={a-bmid a,bin A}} contains an open neighbourhood of the origin.

The general version of the theorem, first proved by André Weil,[2] states that if G is a locally compact group, and A ⊂ G a subset of positive (left) Haar measure, then {displaystyle AA^{-1}={ab^{-1}mid a,bin A}} contains an open neighbourhood of unity.

The theorem can also be extended to nonmeagre sets with the Baire property. The proof of these extensions, sometimes also called Steinhaus theorem, is almost identical to the one below.

Proof The following simple proof can be found in a collection of problems by late professor H.M. Martirosian from the Yerevan State University, Armenia (Russian).

Keeping in mind that for any {displaystyle varepsilon >0} , there exists an open set {displaystyle ,{cal {U}}} , so that {displaystyle Esubset {cal {U}}} and {displaystyle m{cal {U}}

Now assume that {displaystyle |x|

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