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

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

contains an open neighbourhood of the origin.

The general version of the theorem, first proved by André Weil, states that if G is a locally compact group, and A ⊂ G a subset of positive (left) Haar measure, then

$AA^{-1}=\{ab^{-1}\mid a,b\in A\}$

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

$0<\mu (A)<\infty ,$

then for every ε > 0 there are a compact set K and an open set U such that

$K\subset A\subset U,\quad \mu (K)+\epsilon >\mu (A)>\mu (U)-\epsilon .$

For our purpose it is enough to choose K and U such that

$2\mu (K)>\mu (U).$

Since K ⊂ U, for each $k\in K$, there is a neighborhood $W_{k}$ of 0 such that $k+W_{k}\subset U$, and, further, there is a neighborhood $V_{k}$ of 0 such that $2V_{k}\subset W_{k}$. For example, if $W_{k}$ contains $(-\epsilon ,\epsilon )$, we can take $V_{k}=(-\epsilon /2,\epsilon /2)$. The family $\{k+V_{k}:k\in K\}$ is an open cover of K. Since K is compact, one can choose a finite subcover $\{k_{1}+V_{k_{1}},\dots ,k_{n}+V_{k_{n}}\}$. Let $V:=V_{k_{1}}\cap \dots \cap V_{k_{n}}$. Then,

$K+V\subset ((k_{1}+V_{k_{1}})\cup \dots \cup (k_{n}+V_{k_{n}}))+V\subset ((k_{1}+2V_{k_{1}})\cup \dots \cup (k_{n}+2V_{k_{n}}))\subset ((k_{1}+W_{k_{1}})\cup \dots \cup (k_{n}+W_{k_{n}}))\subset U$.

Let v ∈ V, and suppose

$(K+v)\cap K=\varnothing .$

Then,

$2\mu (K)=\mu (K+v)+\mu (K)<\mu (U)$

contradicting our choice of K and U. Hence for all v ∈ V there exist

$k_{1},k_{2}\in K\subset A$

such that

$v+k_{1}=k_{2},$

which means that V ⊂ A − A. Q.E.D.

Corollary

A corollary of this theorem is that any measurable proper subgroup of $(\mathbb {R} ,+)$ is of measure zero.

See also

• Falconer's conjecture

References

• Steinhaus, Hugo (1920), "Sur les distances des points dans les ensembles de mesure positive" (PDF), Fund. Math. (in French), 1: 93–104, doi:10.4064/fm-1-1-93-104.
• Weil, André (1940). L'intégration dans les groupes topologiques et ses applications. Hermann.
• 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. {{cite journal}}: Invalid |ref=harv (help)
• Sadhukhan, Arpan (2020). "An Alternative Proof of Steinhaus's Theorem". American Mathematical Monthly. 127 (4): 330. arXiv:1903.07139. doi:10.1080/00029890.2020.1711693.

• Väth, Martin (2002). Integration theory: a second course. World Scientific. ISBN 981-238-115-5.