Steinhaus theorem

Steinhaus theorem is a theorem in real analysis, first proved by H. Steinhaus, concerning the difference set of a set of positive measure.

The theorem states that if $mu$ is a translation-invariant regular measure defined on the Borel sets of the real line, and $A$ is a Borel measurable set with $mu(A)>0,$ then the "difference set"

: $A-A=\{a-bmid\; a,bin\; A\}$

contains an open neighborhood of the origin. Here, a measure $mu$ is called "translation-invariant" if

:$mu(x+A)=mu(A)$

for all real numbers $x$ and all Borel measurable sets $A$, where $x+A$ is the set of all points of the form $x+a$ with $a$ in $A,$ that is, $x+A$ is obtained by shifting $A$ to the right by $x$.

The theorem extends easily to any Borel-measurable set of positive measure in a locally compact group with identity.

Proof

The following is a simple proof due to Karl Stromberg [http://www.jstor.org/sici?sici=0002-9939(197211)36%3A1%3C308%3ASNAEPO%3E2.0.CO%3B2-L] .

If $mu$ is a regular measure and $A$ is a measurable set, then for every $epsilon\; >0$ there are a compact set $K$ and an open set $U$ such that

:$Ksubset\; A\; subset\; U$ and $mu\; (K)+epsilon>mu(A)>mu(U)-epsilon.$

For our purpose it is enough to choose $K$ and $U$ such that $2mu\; (K)>mu(U).$

Since $Ksubset\; U$, 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 $K+Vsubset\; U$.

Let $vin\; V$ and suppose $(K+v)cap\; K=emptyset.$ Then,

:

contradicting our choice of $K$ and $U$. Hence for all $vin\; V,$ there exist $k\_\{1\},\; k\_\{2\}in\; K\; subset\; A$ such that $v+k\_\{1\}=k\_\{2\},$ which means that $Vsubset\; A-A$. Q.E.D.

References

* cite book
last = Väth
first = Martin,
title = Integration theory: a second course
publisher = World Scientific
date = 2002
pages =
isbn = 9812381155