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\def\Khar{Kharazishvili\xspace}%
\hyphenation{Le-besgue Khar-a-zish-vili 
Kak-u-tani Ox-toby add-i-tive coun-tably
mea-sur-able}
\def\ds{\displaystyle}
\def\sh{{\rm sh}}
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\def\C{\mathbb C}
\def\Q{\mathbb Q}
\def\N{\mathbb N}
\def\T{\mathbb T}
\def\Z{\mathbb Z}
\def\Leb{{\EuScript L}}
\def\K{{\EuScript K}}

\def\c{{\mathfrak c}}
\def\noncont{{\EuScript N}}
\def\B{{\mathfrak B}}
\def\M{{\mathfrak M}}
\def\Hsig{{\EuScript{H}}}
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\def\symdiff{\bigtriangleup}

\def\qq#1{$\ll$#1$\gg$}


\begin{document}

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\begin{center}
RICHARD D. MABRY\footnote{%
Indirizzo dell'Autore: Department of Mathematics, Louisiana State
University in Shreveport, Shreveport, LA 71115-2399, USA.}%

$\overline{~~~~~~~~}$


\renewcommand{\thefootnote}{(**)}%

{\bf \large Some remarks concerning the \\ uniformly gray sets of G.
Jacopini}\footnote%
{Memoria presentata il 21 luglio 1998 da Giorgio Letta, uno dei XL.}%

\bigskip

\begin{minipage}{3.5in}

{\sc Abstract.} --- In a 1995 paper by G. Jacopini, a
$\sigma$-algebra $\Hsig$ of subsets of $\R$ is constructed, and a
translation-invariant measure $\nu$ extending the Lebesgue measure
$\lambda$ is defined on $\Hsig$, such that for each $r\in [0,1]$
there are $E\in\Hsig$ for which $\nu(E\cap A) = r \;\lambda(A)$ for
all Borel subsets $A$. Given these properties, it can be suggested,
as an intuitive interpretation, that such sets as $E$ are
\qq{uniformly gray}.  The main purpose of this note is to discuss the
extent to which such an intuitive characterization is reasonable for
sets
having the above properties. Also mentioned are some other results,
similar to Jacopini's, which have been published elsewhere.
\end{minipage}




\begin{center}
{\bf Alcune osservazioni sugli insiemi \\
uniformemente grigi di G. Jacopini}
\end{center}


\begin{minipage}{3.5in}


{\sc Riassunto.} --- In un articolo del 1995, G. Jacopini esibisce
una
$\sigma$-algebra $\Hsig$ di parti di $\R$ e una misura $\nu$ su di
essa
definita, invariante per le traslazioni, prolungante la misura di
Lebesgue
$\lambda$ e tale che, per ogni numero reale $r$ compreso tra $0$ e
$1$,
esista un elemento $E$ di $\Hsig$ verificante la relazione $\nu(E\cap
A)=r\;\lambda(A)$ per ogni insieme boreliano $A$.  Questa propriet\`a
di
$E$ pu\`o suggerire, dal punto di vista intuitivo, l'i{\nobreak}dea
di un
insieme \qq{uniformemente grigio}.  Lo scopo principale della
presente
nota consiste nel discutere fino a qual punto una simile
interpretazione
intuitiva sia ragionevole, anche alla luce di risultati analoghi a
quello
di Jacopini, pubblicati da altri autori.


\end{minipage}
\end{center}



In the 1995 paper [\ref{Jac}] of G.~Jacopini, the following result,
which we shall call {\sc Theorem~J}, is stated and proved.


\begin{center}
\begin{minipage}{3.6in}

{\sc Theorem:} There exists a $\sigma$-algebra $\Hsig$ of $\R$,
properly containing the Borel $\sigma$-algebra, and a measure $\nu$
on $\Hsig$ extending the Lebesgue measure $\lambda$ and having the
two following properties:


(a) $\nu$ is translation-invariant, in the sense that for each
element $A$ of $\Hsig$ and each real number $t$, one has
$$A+t\in\Hsig \mbox{~~~~and~~~~} \nu(A+t)=\nu (A).$$

(b) For each real number $r$ between 0 and 1, there is an element $E$
of
$\Hsig$ such that for every Borel set $A$ of $\R$, one has
$$\nu(E\cap
A)=r\;\lambda(A).$$

\end{minipage}
\end{center}


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Professor Jacopini offers the intuitive interpretation:
``se si immagina di colorare in nero i punti di $E$, e in bianco
quelli del complementare,''\footnote{English translation: ``if one
imagines coloring the points of $E$ with black and the points of the
complement with white''}
then such sets as $E$ can be thought of as being ``\qq{uniformemente
grigio}''\footnote{``\qq{uniformly gray}''}.


In the 1991 paper [\ref{Mab}], the author of the present note gives
various constructions, some not too dissimilar from the above (see,
e.g., Example 4.7 of that paper), of subsets of $\R$ called {\it
shadings},  and whose definition (below) also suggests \qq{shades of
gray}. (In case the notion of color or shade on a line is not
intuitively appealing, we mention that similar constructions can be
given for subsets of the plane.)


Even though it was not explicitly stated in [\ref{Jac}], we naturally
would hope to associate the number $r$ mentioned in the theorem with
the
\qq{darkness}, \qq{gray-scale} or \qq{shade of gray}, the shades
increasing continuously from white ($r=0$) to black ($r=1$).


What we shall discuss now is the extent to which it may or may not be
reasonable to consider such sets as those with properties (a) and (b)
described in {\sc Theorem~J} as actually having a {\it definite}
shade of
gray in any sense whatsoever. In fact, to the contrary, we note the
existence of $\sigma$-algebras of subsets of $\R$ equipped with
translation-invariant measures, each of which share the properties
(a) and
(b) in {\sc Theorem~J}, but for which it is clearly {\it not}
reasonable
to think of the number $r$ as having anything whatsoever to do with
an
increasing gray-scale as described above. A notable case is found in
the
famous 1950 paper~[\ref{KakOx}] of Kakutani and Oxtoby, in which a
nonseparable extension of the Lebesgue measure is constructed, which
is isometry-invariant and whose character is maximal ($2^\c$). 
(We will use the symbol $\c$ to refer to the cardinality of the
continuum, and also, when convenient, the smallest ordinal having
this
cardinality.) 
To make our point, we need only observe the following, which is a
consequence of the construction in~[\ref{KakOx}].

\bigskip

{\sc Proposition:} {\sl There exists a
$\sigma$-algebra~$\K$ of~$\R$, properly containing the Borel
$\sigma$-algebra, and a family of measures
$\{\nu_\alpha\}_{\alpha<\c}$, each $\nu_\alpha$ defined on $\K$ and
extending the Lebesgue measure $\lambda$, such that the two
following properties hold:


(a) Each $\nu_\alpha$ is translation-invariant, in the sense that
for each element $A$ of $\K$ and each real number $t$, one has
$$A+t\in\K \mbox{~~~~and~~~~} \nu_\alpha(A+t)=\nu_\alpha(A).$$

(b) For every transfinite sequence $(r_\alpha)_{\alpha<\c}$ whose
members are contained in the unit interval $[0,1]$,
there is an element $C$ of $\K$ such that for every Borel set $A$ of
$\R$, one has ({\it for the same} $C$)
$$\nu_\alpha(C\cap A)=r_\alpha\;\lambda(A)~~~\forall \alpha<\c.$$
}

We emphasize that the $\sigma$-algebra $\K$ and the family of
measures $\{\nu_\alpha\}_{\alpha<c}$ are fixed; $C$ depends only upon
the sequence $(r_\alpha)_{\alpha<\c}$. In particular, {\it
all of the $r_\alpha$ may be chosen to be distinct.} So what we have
is a family of translation-invariant extensions of $\lambda$, each
measuring $C$ differently. We sketch the proof of this proposition
below.

In what follows, the cardinality of a set $X$ is denoted by
$\abs{X}$. A subset $X$ of $\R$ will be called  {\it almost
invariant} with respect to a group $G$ of transformations on $\R$
provided that $\abs{X\bigtriangleup g(X)}<\c$, for all $g\in G$.

\bigskip

{\sc Sketch of proof of Proposition:}  The necessary fact is that
there exists a family $\{C_\alpha\}_{\alpha<\c}$ of subsets of $\R$
such that:
\begin{description} 
\item[(1)] $\{C_\alpha\}_{\alpha<\c}$ is a partition of $\R$, i.e., 
$$\R=\biguplus_{\alpha<\c} C_\alpha.$$
(We use $\biguplus$ to emphasize {\it disjoint} unions.)
\item[(2)] Each $C_\alpha$ is
almost invariant relative to the group of all isometries on $\R$.
\item[(3)] If $A$ is a Borel subset of $\R$ and $\lambda(A)>0$, then
$C_\alpha\cap A\not=\emptyset$ for each $\alpha<\c$. 
\end{description}

These properties imply that for each $\alpha<\c$, the set $C_\alpha$
has zero inner measure and full outer measure. (I.e.,
$\lambda_*(C_\alpha)=0$ and $\lambda^*(C_\alpha\cap A)=\lambda(A)$
for each Borel set~$A$. Tus, $C_\alpha$ is 
{\it saturated nonmeasurable}). We let $\Leb$ denote the
Lebesgue measurable sets, ${\cal C}$ the family
$\{C_\alpha\}_{\alpha<c}$, and $\noncont$ the noncontinuum subsets of
$\R$ (i.e., $X\in\noncont$ iff $\abs{X}<\c$). Then the
$\sigma$-algebra $\K$ generated by
${\cal C} \cup\Leb$
is translation-invariant and contains elements of $\noncont$.

Since $\aleph_0\cdot\c=\c,$ we may subdivide the
partition $\cal C$ as follows:
$${\cal C} = \biguplus_{i=1}^\infty\biguplus_{\alpha<c}
C^{(i)}_\alpha.$$
With these properties, if $(x_1, x_2, x_3,\ldots)$
is any sequence whatsoever of nonnegative numbers for which
$\ds\sum_{i=1}^\infty x_i=1$, then measures 
$\{\nu_\alpha\}_{\alpha<\c}$ may be
defined on $\K$  by means of the assignments 
$$\nu_\alpha(X)=0, ~~\forall X\in\noncont\cap\K,$$ 
and 
\begin{equation*}
\nu_\alpha(C^{(i)}_\beta\cap A)= 
\left\{\begin{array}{ll}
x_i \;\lambda(A) & \mbox{ if $\beta=\alpha$}\\
0 & \mbox{ if $\beta\not=\alpha$}
\end{array}\right.
~~~\forall A\in\Leb, ~i=1,2,3,\ldots.
\end{equation*}
Specifically, we shall take 
$$x_i=1/2^i,~~\forall i.$$ 
For each $x\in[0,1]$, let 
$$x = 0.b_1(x)\; b_2(x)\; b_3(x) \cdots 
~~\mbox{(base 2)}$$ be the usual dyadic (nonterminating) expansion of
the number $x$, and let 
$$N(x)=\{i: b_i(x)=1\}.$$ Then 
$$\sum_{i\in N(x)} x_i =x,$$ 
and so, letting 
$$ C=\bigcup_{\alpha<\c}\bigcup\{C^{(i)}_\alpha: i\in
N(r_\alpha)\},$$ 
it is easy to verify that the measures $\{\nu_\alpha\}_{\alpha<\c}$
and the set $C$ will have the desired properties. \hfill$\square$

\bigskip 

Thus, the set $C$ may be considered to have an arbitrarily assigned
(although uniform) shade of gray, with respect to such measures,
including completely black ($r=1$) or white ($r=0$). Given this
ambiguous situation, it is not reasonable to associate with the set
$C$ any definite shade whatsoever.\footnote{One wonders if it is even
reasonable to ask, ``What does such a set really look like?'' for a
set
such as $C$, or indeed, for any of the sets mentioned in this note.}

In connection with this, we should mention that in~[\ref{KhNonsep}],
A.B. \Khar observes this phenomenon of ambiguity, referring also to
the construction in~[\ref{KakOx}], by noting that ``the uniqueness
property [relative to the class of isometry-invariant extensions of
$\lambda$] so characteristic for Lebesgue measure is violated here.''
Briefly, for our purposes, a subset $X$ of $\R$ has the {\it
uniqueness
property} in a class $\M$ of extensions of $\lambda$ provided that
$X$ is
$\mu$-measurable for some $\mu\in\M$ and that whenever $X$ is
$\mu'$-measurable for some $\mu'\in\M$, then $\mu(X)=\mu'(X)$. See,
for
example, the extensive monograph~[\ref{Kh}]~(\S7) or the
paper~[\ref{KhUniq}].

In the same paper~[\ref{KhNonsep}], \Khar then proceeds to construct
a nonseparable isometry-invariant extension of $\lambda$ (having
character $\c$) which does indeed have the uniqueness property in the
class of all isometry-invariant extensions of $\lambda$. In fact,
while
not explicitly stating it in~[\ref{KhNonsep}], the measure
constructed
therein shares the very same properties of the measure $\nu$ as set
forth
in the statement of {\sc Theorem~J}. Having the uniqueness property
in
addition to the aforementioned properties, the sets so obtained might
be
said to have specific, definite shades of gray.

For comparison, let us now suggest another approach to describing
shades of gray. 
Recall that a Banach measure on $\R$ (resp., $\R^2$) is an
isometry-invariant extension of the Lebesgue measure $\lambda$
defined on all subsets of $\R$ (resp., $\R^2$). Such a measure, whose
existence is guaranteed by the Hahn-Banach theorem, is only
finitely-additive. (See the excellent book [\ref{Wag}]~of Stan Wagon
for many details concerning Banach measures.)



{\sc Definition:} (cf.~[\ref{Mab}]) Let $\B$ denote the class of all
Banach measures on $\R$. If $D$ is a subset of $\R$ for which there
is a number $r\in[0,1]$ such that
$$\mu(D\cap A)=r\;\lambda(A)$$
for all bounded Borel sets $A$ and all $\mu\in\B$, then we call $r$
the {\it shade} of $D$, and we write $\sh(D)=r$. We call $D$ an
$r$-shading of $\R$.


Simply put, this definition requires that such a
shading of $\R$ has uniform%
\footnote{%
We should also add that the notion of shade can be generalized as
follows. If $f:\R\to[0,1]$ is a continuous function, then there
exists a subset $D$ of $\R$ for which the {\it shade at each point
$x\in\R$}, denoted by $\sh(D)(x)$, exists and is equal to $f(x)$,
where $\sh(D)(x)$ is  defined by
$\lim_{\delta\to0+}\mu(D\cap(x-\delta,x+\delta))/(2\delta)$, the
values of $\mu(D\cap(x-\delta,x+\delta))$ being independent of
$\mu\in\B$. This is proved in~[\ref{Mab}]. Thus, sets can be {\it
smoothly shaded}. (See~[\ref{KharCar}] for an analogous result
involving the uniqueness property for countably-additive invariant
extensions of $\lambda$.)
}
density and the uniqueness property in the class $\B$.


In spite of the restrictions  associated with finite-additivity,
there are advantages in defining the shade of a set as above, rather
than, say, requiring that sets have uniform density and the
uniqueness property in the class of all {\it countably-additive}
isometry-invariant extensions of $\lambda$. For one thing, the
disadvantage of finite-addivity is greatly  offset by the advantage
of being able to measure all sets!
(For interesting discussions of the relative merits of various
extensions of Lebesgue measure, see the articles
[\ref{AndyJack}]~and~[\ref{Chris}].)
But moreover, consider the following 
subset $W$ of $\R$, which has 
shade equal to zero%
\footnote{%
The set $W$ is an example of a {\it nontrivial homogeneous set} in
the sense of~[\ref{ErdosMarcus}]: $W$ is uncountable, not equal to
$\R$, and $W+w_1-w_2=W$ for all $w_1,w_2\in W$. In that paper it is
proved that every nontrivial homogeneous set has shade equal to zero
(although it is certainly not stated in such terms).
}
([\ref{Mab}], Example 4.2). Let $H$ be a Hamel basis for $\R$
(over $\Q$) and fix any single element $h\in H$. Define $W$ to be the
set
of all $x\in\R$ for which the rational coefficient of $h$ is zero in
the
representation of $x$ with respect to $H$. Then $\R$ is partitioned
into
countably many disjoint translates of $W$:
$$\R=\biguplus_{q\in\Q}(W+qh).$$ As we mentioned, $\sh(W)=0$, which
is
certainly intuitively appealing in view of the observed partition.
But the shade cannot be described in terms 
of countably-additive extensions of $\lambda$, for 
the set $W$ is not even $\nu$-measurable for {\it any}
countably-additive
translation-invariant extension of $\lambda$. (I.e, $W$ is {\it
absolutely
nonmeasurable} in this class of measures. See, e.g.,~[\ref{Kh},~\S4],
[\ref{KhPST},~\S2], or~[\ref{KhAbsNeg}].) To see this, observe that
if
$\nu(W\cap[0,1))=0$, then one obtains the contradiction $\nu(\R)=0$,
because the union of countably many disjoint translates of
$W\cap[0,1)$
can equal all of $\R$. On the other hand, if $\nu(W\cap[0,1))>0$,
then one
obtains the contradiction $\nu([0,1))=\infty$, by observing that for
every $\varepsilon>0$,
infinitely many disjoint translates of $W\cap[0,1)$ exist in
$[0,1+\varepsilon)$.




It must also be noted that in~[\ref{KhPST},~\S8], a construction is
given which has a great deal in common with the construction
in~[\ref{Jac}]. (The construction may also be found
in~[\ref{KhSteinhaus}]. 
Also, see~[\ref{KharCar}] and the famous paper~[\ref{KodKak}] for
invariant extensions of $\lambda$ using many of the same ideas.)

In particular,
on p.~148 of~[\ref{KhPST}],
the $\sigma$-algebra~$S$ of~[\ref{KhPST}] is analagous to the
$\sigma$-algebra~$\Hsig$ of~[\ref{Jac}]. In fact, using the notation
of~[\ref{KhPST}], if one lets $0\le r\le 1$ and
$F=\{x\in\R : (x,f(x))\in\R\times[0,2\pi r)\}$, then the set $F$
satisfies the properties possessed by $E$ as stated in 
{\sc Theorem~J} with the
additional property that $F$ is a Bernstein set. 


Our final remark (which perhaps should have been made earlier!) is
that, despite our having gone to such lengths to point out the
possible ambiguity of the shades of sets having the properties stated
in
{\sc Theorem~J}, it is not difficult to show that the particular
constructions given  for $E$ in~[\ref{Jac}] and $F$ (as above)
in~[\ref{KhPST}] yield sets which {\it do~indeed} have the uniqueness
property (in the class of all translation-invariant extensions
of~$\lambda$).

In fact, the measures on the $\sigma$-algebras $S$ in~[\ref{KhPST}]
and $\Hsig$ in~[\ref{Jac}] are {\it uniquely defined} in the class of
all translation-invariant extensions of $\lambda$ (cf.~[\ref{Kh}]),
which means that all of the elements in 
$S$ and $\Hsig$ have the uniqueness
property. (The same can be said for the measure constructed
in~[\ref{KodKak}], as noted in~[\ref{KharCar}].)
The proof of this uniqueness 
relies upon the essence 
of the final {\sc Remark} given in [\ref{Jac}], 
which we shall put in
the following terms: the measure $\nu$ has the {\it property of
exhaustion}
with respect to translations in $\R$. This means that for 
$X\in\Hsig$ with 
$\nu(X)>0$, there exist $(t_i)_{i\in\N}$ for which
$$\nu\left(\R\setminus
\bigcup_{i\in\N}(X+t_i)\right) = 0.$$ 
(That is, $\nu$ is {\it metrically transitive} with respect to $X$.)
And in the class of
translation-invariant extensions of $\lambda$, the uniqueness
property and
the exhaustion property are equivalent (see~[\ref{KhNonsep}],
Proposition~1).


Just one last thing: the sets $E$ and $F$ are $r$-shadings of $\R$.
Furthermore, this author knows of no subset of $\R$ having uniform
density and the
uniqueness property (relative to all isometry-invariant
countably-additive extensions of $\lambda$) for which $X$ is not also
a shading. 


\begin{center}
{\sc Bibliography}
\end{center}

\newcounter{bibc}
\setcounter{bibc}{0}
\begin{list}{[\arabic{bibc}]}{\usecounter{bibc}}
\item \label{AndyJack}
{\sc A.M. Bruckner} and {\sc J. Ceder}, {On improving Lebesgue
measure},
{\it Normat. Nordisk Matematisk Tidskrift.}, {\bf 23}(1975), 59-68.
\item \label{Chris}
{\sc K. Ciesielski}, {How good is Lebesgue measure?},
{\it Math. Intelligencer}, {\bf 11}(1989), no.~2, 54-58.
\item \label{ErdosMarcus}
{\sc P. Erd\H{o}s} and {\sc S. Marcus},
{Sur la d\'{e}composition de l'espace euclidien en ensembles
homog\`{e}nes},
{\it Acta Math. Hung.}, {\bf 8} (1957), no. 3-4, 443-452.
\item \label{Jac}
{\sc G. Jacopini},
{Una misura invariante per le traslazioni che prolunga la misura di
Lebesgue sulla retta},
{\it Rend. Acc. Naz. Sci. XL, Mem. Mat.}, $113^0$ (1995), XIX,
123-128.
\item \label{KakOx}
{\sc S. Kakutani} and {\sc J. Oxtoby}, 
{Construction of a non-separable invariant extension of the Lebesgue
measure space}, {\it Ann. of Math. (2)}, 
{\bf 52} (1950), no.~3, 580-590.
\item \label{KhNonsep}
{\sc A.B. \Khar}, {On a nonseparable extension of Lebesgue measure},
{\it Russian Acad. Sci. Dokl. Math.}, {\bf 17} (1976), no.~1, 69-72.
\item \label{KhAbsNeg}
{\sc A.B. \Khar}, {On absolutely nonmeasurable and absolutely
negligible sets in classes of invariant measures}, {\it Russian Acad.
Sci. Dokl. Math.}, {\bf 17} (1976), no.~1, 109-112. 
\item \label{KhUniq}
{\sc A.B. \Khar}, Sets with the uniqueness property relative to
Lebesgue 
measure, {\it Russian Acad. Sci. Dokl. Math.}, {\bf 19} (1978),
no.~3, 777-780. 
\item \label{Kh} 
{\sc A.B. \Khar}, {\it Invariant Extensions of
Lebesgue Measure}, Tbilisi Gos. Univ., Tbilisi, 1983 (in Russian).
\item \label{KharCar} 
{\sc A.B. \Khar}, {Some remarks on density points and the uniqueness
property for invariant extensions of the Lebesgue measure}, 
{\it Acta Univ. Carolin. Math. Phys.},
{\bf 35}(1994), no.~2, 33-39.
\item \label{KhSteinhaus} 
{\sc A.B. \Khar}, On the Steinhaus property for
invariant measures, {\it Real Anal. Exchange}, {\bf 21} (1995-1996),
no.~2, 743-749. 
\item \label{KhPST} 
{\sc A.B. \Khar}, {\it Selected Topics of Point Set Theory},
Wydawnictwo Uniwersytetu \L\'{o}dzkiego, \L\'{o}d\'{z}, 1996 
(ISBN 83-7016-929-5).
\item \label{KodKak} 
{\sc K. Kodaira} and {\sc S. Kakutani}, 
{A non-separable translation invariant extension of the Lebesgue
measure space}, {\it Ann. of Math. (2)}, {\bf 52} (1950), no.~3,
574-579.
\item \label{Mab} 
{\sc R. Mabry}, {Sets which are
well-distributed and invariant relative to all isometry-invariant
total extensions of Lebesgue measure}, 
{\it Real Anal. Exchange}, {\bf 16}
(1991-1992), no.~2, 425-459. 
\item \label{Wag} 
{\sc S. Wagon}, {\it The Banach-Tarski Paradox}, Cambridge Univ.
Press, New York, 1985.
\end{list}

\end{document}