Steinhaus theorem

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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 $μ$ is a translation-invariant regular measure defined on the Borel sets of the real line, and $A$ is a Borel measurable set with $μ(A) > 0,$ then the difference set

$A-A=\{a-b\mid a,b\in A\}$

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

μ(x + A) = μ(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.

[edit] Proof

The following is a simple proof due to Karl Stomberg[1].

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

$K\subset A \subset U$ and

μ(K) + ε > μ(A) > μ(U) - ε.

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

Since $K\subset U$ , 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 $K+V\subset U$ .

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

2μ(K) = μ(K + v) + μ(K) < μ(U)

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