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\markboth{{\footnotesize KRZYSZTOF CIESIELSKI}
}{{\footnotesize ISOMETRICALLY INVARIANT  EXTENSIONS}}

\setcounter{page}{799}

\vspace*{-.5in}
\noindent{\footnotesize 
PROCEEDINGS OF THE  \vspace*{-3pt} \\
AMERICAN MATHEMATICAL SOCIETY \vspace*{-3pt} \\
Volume 110, Number 3, November 1990}

\bigskip

\begin{center}
{\large\bf ISOMETRICALLY  INVARIANT  EXTENSIONS\medskip \\
OF LEBESGUE  MEASURE 
}

\bigskip\bigskip

KRZYSZTOF  CIESIELSKI%
{\def\thefootnote{}%
\footnote[1]{Received by the editors October 30, 1989; 
the content of this paper has been presented to Sixth Annual Auburn Miniconference on Real
Analysis on April 7, 1989.}%
\footnote[1]{1980 {\em Mathematics Subject Classification\/} (1985 {\em Revision\/}). Primary
28C10.}% 
}

\bigskip\bigskip

(Communicated by R. Daniel Mauldin)
\end{center}

%Department of Mathematics, West Virginia Univ., Morgantown, WV 26506.


\medskip%\bigskip%\bigskip

  
\begin{quote}{\footnotesize %\small
{\sc Abstract.} The purpose of this note is to give a very short prove of the theorem that every
isometrically invariant measure extending Lebesgue measure on $\real^n$ has a proper isometrically
invariant extension, i.e., that there is no maximal isometrically invariant extension of Lebesgue
measure on  $\real^n$. }
\end{quote}



All the measures we will be considering in this note will be countable additive isometrically
invariant extensions of Lebesgue measure on  $n$-dimensional Euclidean space  $\real^n$ .  By
isometries we will understand bijections of  $\real^n$   that preserves standard Euclidean distance.
All algebraic and measure theoretical terminology used is standard and follows \cite{La,Ru}
respectively.

The first construction of a proper isometrically invariant extension of  Le\-besgue measure
goes back to Szpilrajn's paper \cite{Sz} of 1935.  In the same paper, Szpilrajn stated
Sierpinski's question:  ``Does there exist a maximal isometrically invariant extension of
Lebesgue measure on  $\real^n$''  A negative answer to this question, i.e., the theorem
``every isometrically invariant measure that extends Lebesgue measure on  $\real^n$   has a
proper isometrically invariant extension'' was proved by several mathematicians under
different additional assumptions and restrictions (see
\cite{Pk,Hu,Ha}). Without any assumption the theorem was proved in 1983 by Ciesielski and
Pelc (see
\cite{CP}). For more historical details of this issue see also \cite{Ci1}. The purpose of
this note is to give a very short prove of the theorem that is different from that of
\cite{CP} and follows from the general technique introduced by the author in \cite{Ci2}.

\thm{th}{Let $\mu\colon\M\to[0,\infty]$
be an isometrically invariant extension of Le\-besgue measure on  $\real^n$.  
Then there exists a proper isometrically invariant extension of $\mu$.}

The proof will be based on the following easy and well known Lemmas.

\lem{lem1}{{\rm (Szpilrajn)} Let  $\mu \colon\M\to[0,\infty]$ be an isometrically invariant
measure on 
$\real^n$ .  If a family  $\A$  of subsets of  $\real^n$   is closed under countable union,
close under isometries action (i.e.,  $g[A]\in\A$  for every  $A \in\A$  and every isometry 
$g$)  and such that every  $A\in\A$  has  $\mu$  inner measure zero, then  $\mu$  has an
isometrically invariant extension  $\nu\colon\N\to[0,\infty]$ such that  $\A \subset\N$
and  $\nu(A) = 0$  for every  $A \in  \A$.}

\proof 
If  $\I$  is an ideal of subsets of  $\real^n$   generated by the family  $\A$,  and  $\N$ 
stands for a $\sigma$-algebra generated by  $\M \cup   \I$  then all elements of  $\N$  are
of the form  $(M  \cup   I_1) \setminus  I_2$  where  $M \in  \M$  and  $I_1,I_2 \in  \I$. 
It is easy to see that  $\nu \colon \N \to  [0,\infty]$  such that  $\nu((M  \cup    I_1) 
\setminus   I_2) = \mu(M)$ is a well defined isometrically invariant measure on  $\real^n$  
extending  $\mu$.

\medskip

In the proof of the next lemma we use a method which goes back to Harazis\-vili's paper
\cite{Ha} (see also \cite{Ci2}).

\lem{lem2}{Let  $\real^n=\bigcup\{N_k\colon k=0,1,2,\ldots\}$. If each $N_k$  satisfies the
condition
\begin{quote}
for every countable set  $\{g_r\colon r =0,1,2,\ldots\}$ 
of isometries there 
is an uncountable set $H$ of isometries such that for every distinct 
$h_1,h_2\in H$
\end{quote}
\vspace*{-9pt}
{\rm ($*$)}
\vspace*{-33pt}
\begin{quote}
\begin{multline*}
\hspace*{45pt}h_1(\bigcup\{g_r[N_k]\colon r=0,1,2,\ldots\}) \\
\cap h_2(\bigcup\{ g_r[N_k]\colon r=0,1,2,\ldots\})=\emptyset\hspace*{45pt}
\end{multline*}
\end{quote}
then every isometrically invariant extension $\mu\colon\M\to[0,\infty]$ of Lebesgue measure
on  $\real^n$  has a proper isometrically invariant extension.
}

\proof 
Let  $\mu\colon\M\to[0,\infty]$  be an isometrically invariant extension of Lebesgue measure
on  $\real^n$.  Define
$$\A_k= \left\{\bigcup \{g_r[N_k]\colon r=0,1,2,\ldots \} 
\colon  \mbox{where all $g_r$'s are isometries of $\real^n$}\right\}.$$


If  $M \in  \M$  is a subset of  $A \in  \A_k$  then  $h_1[M]\cap h_2[M] =
\emptyset$   for every distinct  $h_1,h_2$  from  $H$.  But  $\mu(h[M]) =
\mu(M)$  for every  $h$  from  $H$.  Moreover measure  $\mu$  is 
$\sigma$-finite as an extension of Lebesgue measure. This implies that  $\mu(M)
= 0$  and so  A  has  $\mu$  inner measure zero. 

Thus we proved that every 
$\A_k$  satisfies assumptions of Lemma 1. Hence  for each  $k = 0,1,2,\ldots$  
there is an isometrically invariant extension  $\nu_k$  of  $\mu$  such that 
$\nu_k(N_k) = 0$.  But all  $N_k$'s  cannot have  $\mu$  measure zero. So some 
$\nu_k$  must be a proper extension of $\mu$.

\medskip


The following lemma is an elementary geometrical fact and will be left without
the proof.

\lem{lem3}{Every isometry of  $\real^n$  can be represented as a superposition  
$t\circ L$ where  $t$ is a translation by a vector $(t_1,t_2,\ldots,t_n)$  and 
$L$  is a linear transformation of  $\real^n$  represented by some  $n\times
n$  matrix  $(a_{ij})$. }

\noindent{\em Proof of the Theorem.}  By Lemma 2 it is enough to construct 
$N_k$'s  such that  $\real^n=\bigcup \{N_k\colon k=0,1,2,\ldots \}$   and each 
$N_k$  satisfies condition ($*$).

Let  $\B$  be a transcendence base of  {\real}  over  {\rational}  and let us
represent  $\B$  as  $\B = \bigcup \{\B_k\colon k=0,1,2,\ldots \}$
   where  $\B_0\subset\B_1\subset\B_2\subset\cdots$   and  $\B_{k+1}\setminus
\B_k$  is uncountable. 
Define
\[
N_k=[\cl_{\real}({\rational}(\B_k))]^n
\]
where  ${\rational}(\B_k)$  is a field generated by  ${\rational}$  and 
$\B_k$  and  $\cl_{\real}({\rational}(\B_k))$  is an algebraic  closure of 
${\rational}(\B_k)$  in  ${\real}$.  We have to prove that  $N_k$'s satisfy 
($*$).

So let us choose  $k$  and a countable set  $\{g_r\colon  r = 0,1,2,\ldots
\}$   of isometries. There exists a countable set  $\A \subset \B$  such that
all  $g_r$'s are defined over  $\cl_{\real}({\rational}(\A ))$, i.e., that for
each  $g_r$  the coefficients  $t_i$'s  and  $a_{ij}$'s  from Lemma 3 are in 
$\cl_{\real}({\rational}(\A))$. Let  $L=\cl_{\real}({\rational}(\A \cup 
\B_k))$.  Then
\[
\bigcup \{ g_r[N_k]\colon r=0,1,2,\ldots \} \subset L^n.
\]

Define
\[ 
H=\{ t_\alpha\colon \alpha\in \B_{k+1}\setminus (\A \cup  \B_{k})\},
\]  
where  $t_\alpha$  is a translation by a vector  $(\alpha,0,0,\ldots ,0)$. 
Then  $H$  is uncountable  and for distinct $\alpha,\beta\in H$
\[ 
t_\alpha(\bigcup \{ g_r[N_k]\colon r=0,1,2,\ldots \} ) \cap   
t_\beta(\bigcup \{ g_r[N_k]\colon r=0,1,2,\ldots \} )
\]
\vspace*{-24pt}
\[
\subset  t_\alpha(L^n)
\cap    t_\beta(L^n) = \emptyset  
\] 
as  $\alpha-\beta\notin  L$.  This finishes the proof of the theorem.

{\footnotesize
\begin{thebibliography}{222}

\bibitem[Ci1]{Ci1} K. Ciesielski, {\em How good is Lebesgue measure?}, Math. Inteligencer {\bf }
(1989), 54--58.

\bibitem[Ci2]{Ci2} K. Ciesielski, {\em Algebraically invariant extensions of $\sigma$-finite
measures on Euclidean space}, Trans. Amer. Math. Soc. {\bf 315} (1989).

\bibitem[CP]{CP}	K. Ciesielski and  A. Pelc, {\em Extensions of invariant measures on Eucledean
spaces}, Fund. Math. {\bf 125} (1985), 1--10.

\bibitem[Ha]{Ha} 	A. B. Harazisvili, {\em On Sierpinski's problem concerning strict 
extendibility of an invariant measure}, Soviet Math. Dokl. {\bf 81} (1977), 71--74.

\bibitem[Hu]{Hu}	A. Hulanicki, {\em Invariant extensions of the Lebesgue measure}, Fund. Math.
{\bf 51} (1962), 111--115.

\bibitem[La]{La}	S. Lang, {\em Algebra}, Addison-Wesley, 1984.

\bibitem[Pk]{Pk}	S. S. Pkhakadze, {\em K teorii lebegovskoi miery}, Trudy Tbiliss. Mat.
Inst. Razmadze Acad. Nauk Gruzin. SSR, vol. 25, 1958. (Russian)

\bibitem[Ru]{Ru}	W. Rudin, {\em Real and Complex Analysis}, McGraw-Hill, 1987.

\bibitem[Sz]{Sz}	E. Szpilrajn, {\em Sur l'extension de la measure lebesguienne}, Fund. Math.
{\bf 25} (1935), 551--558. (French)
\end{thebibliography}

%\vspace*{-.1in}

 {\sc Department of Mathematics, West Virginia University, Morgantown, 
West Virginia 26506}}

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