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[edit]

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[edit]

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

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 , there is a neighborhood of 0 such that , and, further, there is a neighborhood of 0 such that . For example, if contains , we can take . The family is an open cover of K. K is compact, hence one can choose a finite subcover . Let . 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.

Consequence[edit]

A consequence is, that any measurable proper subgroup of (R,+) is of measure zero.

See also[edit]

Notes[edit]

References[edit]