Steinhaus theorem

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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.[1]

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.

More generally, if G is a locally compact group, and A ⊂ G is 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.[2] 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, 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 ⊂ 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.

See also

Notes

  1. ↑ Steinhaus (1920); Väth (2002)
  2. ↑ Stromberg (1972)

References

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