\documentclass%[12pt] {article} \usepackage{amssymb} \usepackage{amsmath} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newcommand{\thm}[2]{\begin{theorem}\label{#1}#2\end{theorem}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}#2\end{lemma}} \def\proof{\noindent {\em Proof. }} \def\M{{\mathcal M}} \def\N{{\cal N}} \def\I{{\cal I}} \def\A{{\cal A}} \def\B{{\cal B}} \def\cl{{\rm cl}} \def\real{{\bf R}} \def\rational{{\bf Q}} \def\natural{{\bf N}} \begin{document} \thispagestyle{plain} \pagestyle{myheadings} \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}} \end{document}