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, for each , there is a neighborhood of 0 such that , and, further, there is a neighborhood of 0 such that . For example, if contains , we can take . The family is an open cover of K. K is compact, hence one can choose a finite subcover . Let . Then,
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.
A consequence is, that any measurable proper subgroup of (R,+) is of measure zero.
- 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. doi:10.2307/2039082. JSTOR 2039082.