Steinhaus theorem

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 (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 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 k ∈ K , there is a neighborhood W k of 0 such that k + W k ⊂ U , and, further, there is a neighborhood V k of 0 such that 2 V k ⊂ W k . For example, if W k contains ( − ϵ , ϵ ) , we can take V k = ( − ϵ / 2 , ϵ / 2 ) . The family { k + V k : k ∈ K } is an open cover of K. K is compact, hence one can choose a finite subcover { k 1 + V k 1 , … , k n + V k n } . Let V := V k 1 ∩ ⋯ ∩ V k n . 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 ( R , + ) is of measure zero.