Steinhaus theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 in real analysis

Real analysis

Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

, first proved by H. Steinhaus, concerning the difference set of a set of positive measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

.

The theorem states that if is a translation-invariant regular measure

Regular measure

In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...

  defined on the Borel set

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...

s of the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, and is a Borel measurable set with then the difference set

contains an open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 neighborhood of the origin. Here, a measure is called translation-invariant if


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

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

Locally compact group

In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...

.

Analogous theorem for nonmeagre

Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...

 sets with Baire property also holds, and is sometimes called Steinhaus theorem, proof of which is almost identical to the one below.

Proof

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

If is a regular measure and is a measurable set with positive finite measure , then for every there are a compact set  and an open set such that
and

For our purpose it is enough to choose and such that

Since , there is an open cover of that is contained in . is compact, hence one can choose a small neighborhood of such that

Let and suppose Then,


contradicting our choice of and . Hence for all there exist such that
which means that . Q.E.D.

Q.E.D.

Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...