Steinhaus theorem
From Wikipedia, the free encyclopedia
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
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
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 
, 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 
.
Let 
and suppose 
Then,
- 2μ(K) = μ(K + v) + μ(K) < μ(U)
contradicting our choice of K and U. Hence for all 
there exist 
such that v + k1 = k2, which means that 
. Q.E.D.
[edit] References
- Väth, Martin, (2002). Integration theory: a second course. World Scientific. ISBN 9812381155.
