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.
contains an open neighbourhood of the origin.
contains an open neighbourhood of unity.
The following is a simple proof due to Karl Stromberg. If μ is the Lebesgue measure and A is a measurable set with positive finite measure
then for every ε > 0 there are a compact set K and an open set U such that
For our purpose it is enough to choose K and U such that
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.
Let v ∈ V, and suppose
contradicting our choice of K and U. Hence for all v ∈ V there exist
which means that V ⊂ A − A. Q.E.D.
- Steinhaus, Hugo (1920), "Sur les distances des points dans les ensembles de mesure positive" (PDF), Fund. Math. (in French) 1: 93–104.
- Stromberg, K. (1972). "An Elementary Proof of Steinhaus's Theorem". Proceedings of the American Mathematical Society 36 (1): 308. JSTOR 2039082.