Set of stone house

The set of stone house is a theorem of the mathematical part of the area of the measure theory , the work of a Polish mathematician Hugo Steinhaus of the first volume Fundamenta Mathematicae back (1920). It deals with a fundamental topological property of the Lebesgue-measurable subsets of the -dimensional real coordinate space . ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

Formulation of the sentence

Steinhaus's theorem says:

Forming a Lebesgue measurable subset with Lebesgue measure the amount of all of two elements from formable differences , so is thereby given amount is always an environment of . ${\displaystyle A\subseteq \mathbb {R} ^{n}\;(n=1,2,3,\ldots )}$ ${\displaystyle \lambda ^{n}(A)>0}$ ${\displaystyle A}$ ${\displaystyle A-A=\{a_{1}-a_{2}\;\colon \;a_{1}\in A\;,\;a_{2}\in A\}}$ ${\displaystyle \mathbf {0} }$

In other words:

Under the conditions mentioned, there is always an open full sphere . ${\displaystyle U_{\delta }(\mathbf {0} )\subseteq A-A\;(\delta >0)}$

Conclusions: Two sentences by Sierpiński

To the set of stone house two sets of the Polish mathematician may Wacław Sierpiński about Hamel bases of a vector space over the field of rational numbers are returned. They can be specified as follows: ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$

Let a Hamel basis of the vector space be given . ${\displaystyle B}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

Then:

(1) If Lebesgue is measurable in , then is a Lebesgue zero set , i.e. of the Lebesgue measure . ${\displaystyle B}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle B}$ ${\displaystyle \lambda ^{1}(B)=0}$

(2) If a non-empty and at most countable subset of and is the - linear envelope of , then is a non-Lebesgue-measurable subset of . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle M_{A}:={\operatorname {span} }_{\mathbb {Q} }(B\setminus A)}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle B\setminus A}$ ${\displaystyle M_{A}}$ ${\displaystyle \mathbb {R} }$

To prove the two inferences

A proof for (1) can then be given to Jürgen Elstrodt as follows:

If such a Hamel basis is assumed to be Lebesgue measurable with Lebesgue measure , a contradiction arises . ${\displaystyle B}$ ${\displaystyle \lambda ^{1}(B)>0}$

Since such a Hamel basis is not the empty set , one can be selected and thus the real zero sequence can be formed. ${\displaystyle B}$ ${\displaystyle a\in B}$ ${\displaystyle \left({\tfrac {a}{\nu }}\right)_{\nu \in \mathbb {N} }}$

Now comes to fruition, then that but after the set of stone house must be a neighborhood of zero, which is why almost all members of the zero sequence contained therein must be. ${\displaystyle B-B}$

So there is also a natural number and two different ones for the ${\displaystyle n}$ ${\displaystyle b,c\in B}$

${\displaystyle {\frac {1}{n}}\cdot a=b-c}$

applies.

But that means that too

${\displaystyle 0={\frac {1}{n}}\cdot a+(-1)\cdot b+1\cdot c}$

applies.

This one has a nontrivial representation of a linear combination of elements from with coefficients from what is inconsistent with the requirement that a Hamel basis of more than should be. ${\displaystyle 0\in \mathbb {R} }$ ${\displaystyle B}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$

Therefore, such a Lebesgue-measurable Hamel basis can only be a Lebesgue zero set. ${\displaystyle B}$

The proof of (2) is similar and is based on the translational invariance of the Lebesgue measure and the fact that it always holds. ${\displaystyle M_{A}-M_{A}=M_{A}}$

Remarks

• According to Jürgen Elstrodt, the sentence reinforces the intuitive idea that every Lebesgue-measurable subset of is approximately equal to an open set. Here it is even true that the Lebesgue-measurable subsets of have the following characteristic property : ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$

To a predetermined barrier there in always a open set and a closed set with ${\displaystyle \delta >0}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle U}$ ${\displaystyle F}$

${\displaystyle F\subseteq A\subseteq U}$and . ${\displaystyle \lambda ^{n}(U\setminus F)<\delta }$

• How to that of Karl Stromberg in the Proceedings of the American Mathematical Society takes 1972 delivered note, there is the sentence a generalization to locally compact groups with haarschem measure , which also set Steinhaus ( English Steinhaus theorem is) called and their formulation the French mathematician André Weil goes back.
• Far more is known about the Hamel bases from over . For example, it can be shown that such a Hamel base can never be a Borel set of . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} }$

swell

• Jürgen Elstrodt: Measure and integration theory (=  Springer textbook - basic knowledge of mathematics ). 7th, corrected and updated edition. Springer-Verlag , Heidelberg (inter alia) 2011, ISBN 978-3-642-17904-4 .
• Hugo Steinhaus: Sur les distances des points dans les ensembles de mesure positive . In: Fundamenta Mathematicae . tape 1 , 1920, p. 93-104 ( online [PDF]).
• Karl Stromberg: An Elementary Proof of Steinhaus's Theorem . In: Proceedings of the American Mathematical Society . tape 36 , 1972, p. 308 , JSTOR : 2039082 ( MR0308368 ).

Individual evidence

1. ↑ ^{a b c} Jürgen Elstrodt: Measure and integration theory. 2011, pp. 67-68
2. ↑ ^{a b c} Jürgen Elstrodt: Measure and integration theory. 2011, pp. 99-100