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.
More generally, if G is a locally compact group, and A ⊂ G is a subset of positive 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 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.
Since K is compact, one can choose a finite subcover.
Let. Then,
Let v ∈ V, and suppose
Then,
contradicting our choice of K and U. Hence for all v ∈ V there exist
such that
which means that V ⊂ A − A. Q.E.D.

Corollary

A corollary of this theorem is that any measurable proper subgroup of is of measure zero.