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. 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" contains an open neighbourhood of the origin. The general version of the theorem, first proved by André Weil, states that if "G" is a locally compact group, and "A" ⊂ "G" a subset of positive (left) Haar measure , then 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 formula_3, there exists an open set formula_4, so that formula_5 and formula_6, for a given formula_7, we can find an appropriate interval formula_8 so that taking just an appropriate part of positive  measure of the set formula_9 we can assume that formula_10, and that formula_11. Now assume that formula_12, where formula_13. We'll show that there are common points in the sets formula_14 and formula_9. Otherwise formula_16. But since formula_17, when formula_12, it follows immediately that formula_19, what we needed to establish. Corollary. A corollary of this theorem is that any measurable proper subgroup of formula_20 is of measure zero.