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\documentclass{rae}
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%\coverauthor{Brian S. Thomson}
%\covertitle{Some Properties of Variational Measures}
\received{August 20, 1998}
\MathReviews{26A45, 26A39, 28A12}
\firstpagenumber{1}
\markboth{Brian S. Thomson}{Variational Measures}
\author{Brian S. Thomson,Mathematics Department,Simon Fraser University,
B.~C., Canada V5A~1S6.
e-mail: {\tt thomson@cs.sfu.ca}}
\title{SOME PROPERTIES OF VARIATIONAL MEASURES}
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\begin{document}
\maketitle
\begin{center}
{\em Dedicated to the memory of
Vasile Ene (1957--1998).}
\end{center}
\begin{abstract}
Recently several authors have established
a remarkable property of the variational
measures associated with a function.
Expressed in classical language, this property
asserts that if a function is ACG${}_*$ on
all sets of Lebesgue measure zero then the function
must be globally ACG${}_*$.
This article is an exposition of
some ideas related to this property
with the intention of bringing it to the attention of a wider
audience than these original papers might attract.
\end{abstract}
Recent years have seen continued interest
in the variational measures associated with
a function, e.g.,
\cite{bened},
\cite{bened2}, \cite{bened2a}, \cite{bened3},
\cite{ZoliWash2}, \cite{ZoliWash}, \cite{ZoliWash3},
\cite{Ene1}, \cite{Ene39}, \cite{ene1},
\cite{Pfeffer}, \cite{Pfeffer3},
\cite{Pfeffer4},
\cite{thomson1}, and \cite{thomson2}.
In the simplest setting a function $f:[a,b]\to\Reals$
is given and one constructs a measure $\mu_f$ that
carries the variational information about $f$. If $f$
is of bounded variation then $\mu_f$ is the usual
Lebesgue-Stieltjes measure associated with the total
variation function of $f$. In general a measure $\mu_f$
can be constructed for arbitrary functions and which
has considerable power to express properties of $f$.
Perhaps the nicest elementary uses of this measure would
be in the following assertions.
\begin{quote}
{\em
If $f:[a,b]\to\Reals$ then a necessary and sufficient condition
for the identity $f(x)-f(a)= \int_a^x f'(t)\,dt$
in the sense of the Lebesgue integral is that $\mu_f$
is finite and absolutely continuous with respect to
Lebesgue measure on $[a,b]$.
}
\end{quote}
\begin{quote}
{\em
If $f:[a,b]\to\Reals$ then a necessary and sufficient condition
for the identity
$f(x)-f(a)= \int_a^x f'(t)\,dt$
in the sense of the Denjoy-Perron integral is that $\mu_f$
is $\sigma$--finite and absolutely continuous with respect to
Lebesgue measure on $[a,b]$.
}
\end{quote}
W.~F.~Pfeffer, with characteristic insight, conjectured in 1994 that
in the latter assertion the assumption that $\mu_f$
is $\sigma$--finite may be dropped, in fact that the property
that $\mu_f$
is absolutely continuous with respect to
Lebesgue measure on $[a,b]$ is already enough to deduce that
it is also $\sigma$--finite. This remarkable property of
the variational measure has since been proved, both on the
real line
(\cite{bened2}, \cite{bened2a}, \cite{Ene39})
and in various higher dimensional versions
(\cite{bened3}, \cite{ZoliWash2}, \cite{ZoliWash}).
It is this property that we propose to study in this short
article.
The property can be expressed directly, too, in the more classical
language familiar to most real analysts. Roughly it asserts that
to test that a function is
ACG${}_*$
on a set $E$ it would be enough to test that it is
ACG${}_*$
on measure zero subsets of $E$ (cf.~Ene~\cite[p.~58]{Ene1}).
To better appreciate the
surprising feature of this observation we should note
that it was entirely overlooked
by Denjoy and Saks
who, most of us surely felt,
had exhausted the study of the VBG${}_*$
and ACG${}_*$ classes of functions.
Since the proof does not require techniques with which
they were unfamiliar it was only that this property
did not occur to them.
It is, by no means, the case that all Borel measures on the interval
$[a,b]$ would have the Pfeffer property. For example,
simply take $\mu(B)=0$
for all Borel sets of measure zero and $\mu(B)=\infty$ for
the remaining Borel sets.
What is there about the variational measures that allows this
feature, that the behavior on the measure zero Borel sets
imposes some global behavior?
Since the proof in~\cite{bened2} uses the language of ACG${}_*$
functions and that in~\cite{ene1} uses the language of VBG${}_*$
functions
it may not be immediately clear that this is a feature of the
method used to construct the measures and not a property
merely of functions.
The measure arguments in~\cite{bened3},~\cite{ZoliWash} and~\cite{ZoliWash3}
require different techniques since they address the problem in
higher dimensions.
The simple technique used here is adapted from~\cite{bened2a}.
\bigskip
\noindent
{\bf \S 1.}
Let us begin by
recalling the method (sometimes known as Method~III)
defining the measure associated
with any nonnegative interval function $\tau$ on $[a,b]$.
Let $E \subset [a,b]$, let $\delta$ be a gauge on $E$
(i.e., $\delta$ is a positive function defined on $E$)
and write
$$
V(\tau,E,\delta)
=
\sup \left\{
\sum \tau(a_i,b_i)
\right\}
,$$
where the supremum is taken over all
disjoint collections $\{(a_i,b_i)\}$
of open
subintervals of $(a,b)$
for which there is a point $\xi_i\in E\cap (a_i,b_i)$
satisfying $b_i-a_i< \delta(\xi_i)$.
Then write
$$
\mu_\tau^*(E) = \inf \left\{ V(\tau ,E,\delta):
\text{ $\delta$ is a gauge on $E$}
\right\}
.
$$
It can be verified that $\mu_\tau^*$ is a metric outer measure
on $[a,b]$. Since it is a metric outer measure its restriction
to the Borel sets is a measure $\mu_\tau $; we call $\mu_\tau$ the
{\em Method~III measure} associated
with the interval function $\tau$.
If $f$ is continuous and monotonic and $\tau(a,b)= |f(b)-f(a)|$
then $\mu_\tau$ is precisely the Lebesgue-Stieltjes measure generated
by $f$. If $f$ is continuous and has bounded variation then
$\mu_\tau$ is the Lebesgue-Stieltjes measure associated with the
total variation function for $f$. (Accounts of metric outer
measures can be found in numerous texts, for example in
Bruckner et al.~\cite{BBT}
and Edgar~\cite{edgar} or~\cite{edgar2}
where also this method of construction is
discussed.)
We show that all measures constructed in this
manner have the Pfeffer property,
in fact that if $\mu_\tau(B)$ is $\sigma$--finite for all
Lebesgue
measure zero closed sets $F \subset E$ where $E$ is closed then $\mu_\tau$
is $\sigma$--finite on $ E$.
In particular it follows that if $\mu_\tau$ vanishes on
all
Lebesgue
measure zero subsets of $ E$
then
$\mu_\tau$
is $\sigma$--finite on $ E$.
The method of proof uses a standard Baire category argument and
a clever construction of a measure zero Cantor set in $E$;
it is adapted from~\cite{bened2} where it is used in the setting
of ACG${}_*$ functions (cf.~also Pfeffer~\cite[pp.~7-8]{Pfeffer3}).
\smallskip\smallskip
\noindent
{\bf THEOREM~1.}
%%\begin{thm}
{\em
%\begin{theorem}
Let $\mu_\tau$ be a measure constructed
from
an interval function $\tau$ and let $E \subset [a,b]$
be closed. If $\mu_\tau$ is $\sigma$--finite on all
closed subsets of $E$ that have zero Lebesgue measure, then
$\mu_\tau$ is $\sigma$--finite on $E$.
}
%%\end{thm}
\smallskip
\noindent {\bf Proof.}
%%%\end{theorem}
%%%\begin{proof}
Let $P$ be the set of all points $x\in E$ for which
$\mu_\tau$ is non $\sigma$--finite on $E \cap (c,d)$ for
every interval $(c,d)$
containing $x$.
We claim that $P$ is empty. If so the theorem is easy
to prove. Associated with every point $x$ in the compact
set $E$ is an open interval $I_x$ so that $\mu_\tau$ is
$\sigma$--finite on $E \cap I_x$. A compactness argument
reduces this open cover to a finite subcover and
shows that $\nu_\tau$ is
$\sigma$--finite on $E $.
Let us show that $P$ is empty by obtaining a contradiction
from the assumption that $P \not = \emptyset$.
If $P$ is nonempty then it is a perfect subset of $[a,b]$.
Clearly $P$ is closed. We claim that it has no isolated points.
To
see this suppose that $x_0$ is, if possible, an isolated
point of $P$. Then, since $\{x_0\} \cap E$ is measure
zero trivially,
$\mu_\tau$ is
$\sigma$--finite on $\{x_0\} \cap E$
(i.e., it is finite).
Because $x_0$ is an isolated
point of $P$ it follows that $\mu_\tau$ is
also $\sigma$--finite
on
$[a_i,a_{i+1}] \cap E$ for some sequence
of points $a_i \nearrow a$ and it is $\sigma$--finite
on
$[b_{i+1}, b_i] \cap E$ for some sequence
of points $b_i \searrow a$.
It follows, then that $\mu_\tau$ is
$\sigma$--finite on $[a_1,b_1] \cap E$
which means that $x_0$ could not have been a point of $P$.
Continuing to assume that $P \not=\emptyset$, we
choose a finite, disjoint collection
$$
I_{11}, I_{12}, I_{13}, \dots
$$
of open subintervals of $(a,b)$
(at least three such intervals in any case)
so that each contains a point of $P$ and so that
$$
\sum_j |I_{1j}| < 1/2
$$
(here $|I|$ is used to denote the length of an interval
$I$) and
$$
\sum_j \tau(I_{1j}) >2
.
$$
The reason we can do this is that
if $J$ is any interval containing a point of $P$
then $V(\tau,P \cap J,\delta)=\infty$
for any $\delta$.
For if not then $\mu_\tau(P \cap J)<\infty$
and $\mu_\tau$ is $\sigma$--finite on
each set $E \cap (c_i,d_i)$
where $\{(c_i,d_i)\}$
is the sequence of intervals complementary to
$P$ in $J$. But
$$
E \cap J = (P \cap J) \cup \bigcup_i
\left( E \cap (c_i,d_i) \right)
$$
and it would follow that $\mu_\tau$ is $\sigma$--finite on
$E \cap J $ which is not possible if
$J$ contains a point of $P$.
Since $V(\tau,P \cap J,\delta)=\infty$
for any $\delta$,
a disjoint sequence of
subintervals $I_1$, $I_2$, $I_3$, \dots of $J$
each containing a point of $P$ can be selected with
$$ \sum_k \tau(I_k)$$
as large as we please.
Thus, since $P$ is perfect we can begin by selecting three disjoint
intervals $J$, $J'$, and $J''$ each containing a point of $P$,
and each with length less than $1/6$,
and find inside them enough further open subintervals
to provide the sequence $I_{11}$, $I_{12}$, $I_{13}$, \dots
containing at least three members and with the desired properties.
Now inside each interval $I_{11}$, $I_{12}$, $I_{13}$, \dots
we can apply the same argument to find still smaller intervals.
Let us set $I_{01}=(a,b)$ and proceed inductively
using the same argument at each stage.
We construct
disjoint intervals
$$
I_{i1}, I_{i2}, I_{i3}, \dots
$$
so that
\begin{enumerate}
\item
each $I_{ij}$ is an open subinterval of some previous
level interval $I_{(i-1)k}$,
\item
each interval contains a point of $P$,
\item
each
such interval $I_{(i-1)k}$ contains at the next
level at least three such intervals $I_{ij}$
and
\item
for any $i=1,2,3, \dots$
$$
\sum_j |I_{ij}| < 2^{-i}
$$
and, finally
\item
for any $i=1,2,3, \dots$
and if there is an interval $I_{(i-1)k}$ then
$$
\sum_j \{ \tau(I_{ij}) :
I_{ij} \subset I_{(i-1)k}
\}
> 2^i
.
$$
\end{enumerate}
Define now the set
$$
N = E \cap \bigcap_i \bigcup_j \overline{I_{ij}}
.
$$
This set $N$ is closed. Compactness of the sets
ensures that it is nonempty.
It is of Lebesgue
measure zero because of the requirement~(e) in the construction
of the intervals.
Since $N$ is a closed, measure zero subset of $E$ the measure
$\mu_\tau$ must be, by hypothesis,
$\sigma$--finite on $N$.
Let $N_1$, $N_2$, $N_3$, be a sequence of disjoint Borel
subsets of
$N$ on each of
which $\mu_\tau$ is finite and whose union is all of $N$.
Choose a gauge $\delta$ on $N$ so that
$$
V(\tau,N_p, \delta)
< \infty
$$
for each $p=1,2,3, \dots$.
Let $$E_m= \{ x\in N: \delta(x)> 1/m \}$$
for each $m=1,2,3, \dots$.
Note that $E_m \nearrow N$. Thus the sets
$\{E_m \cap N_p\}$ for $m=1,2,3, \dots$
and $ p=1,2,3, \dots$
form a countable cover of $N$.
By the Baire category theorem there is an open interval $I$
and a member $N_p \cap E_m$ of the cover
so that $N_p\cap E_m$ is dense in the nonempty portion $N \cap I$.
By passing to a subinterval if necessary,
we can assume that $|I|< 1/m$.
On the one hand, we have from the way in which we constructed
the gauge
\begin{equation}\label{2}
V(\tau, E_m \cap N_p, \delta)
\leq V(\tau,N_p , \delta)
< \infty
.
\end{equation}
But, on the other hand,
since $I$ contains points of $N$ there must be for all sufficiently
large $i$ some $k$ so that $I_{(i-1)k} \subset I$. Each interval
$$I_{ij} \subset I_{(i-1)k}$$ must contain a point of $N$;
since $N_p\cap E_m$ is dense in the portion $N \cap I$
each such interval also contains a point $\xi$ of $N_p\cap E_m$.
But
the length of such an interval would be smaller
than $I$ which is smaller than $1/m$ which is smaller
than $\delta(\xi)$.
Consequently from the requirement~(e)
$$
2^ i < \sum_j \{ \tau(I_{ij}) :
I_{ij} \subset I_{(i-1)k}
\}
\leq
V(\tau, E_m \cap N_p , \delta)
.
$$
This would be valid for all sufficiently large $i$ and that is impossible
because of~(\ref{2}).
Thus
we have reached
a contradiction and completed the proof.
\nobreak \hfill
$\blacksquare$
\smallskip
To express our theorem in another way we could observe
that if $\mu_\tau$ is non $\sigma$--finite on
a closed set $E$ then $\mu_\tau$ is non $\sigma$--finite on
many closed null subsets of $E$. How many? Can one say
anything about the subsets of $E$ of Hausdorff dimension
smaller than $1$? For answers to these questions
(and others) in $\Reals^n$
for any dimension $n$ see the interesting papers
of B.~Bongiorno et al.~\cite{bened3}
and
Z.~Buczolich and W.~F.~Pfeffer~\cite{ZoliWash2}.
\bigskip
\noindent
{\bf \S 2.}
We can take another perspective on the result in Section~1.
Implicit in this method of constructing a measure is a
differentiation basis and the derivates of the interval
function $\tau$ play an important role in studying the
measure $\mu_\tau$. (For a deeper account of this
role see~\cite{thomson1}.)
Define for any $x \in (a,b)$,
the upper derivate of $\tau$ at the point $x$,
$
\overline{D} \tau(x)
$,
to be
$$
\inf_{\delta>0}
\sup
\left\{
\frac{\tau(I)}{|I|}:
\text{ $I$ an open subinterval of $(a,b)$ with
$x \in I$ and $|I|<\delta$}
\right\}
.
$$
The
lemma shows that the $\sigma$--finiteness of the
measure $\mu_\tau$,
which was our concern in the preceding section,
has a great deal to do with this
derivate.
\bigskip
\noindent
{\bf LEMMA.}
{\em
Let $\tau$ be an arbitrary nonnegative interval function
and $\mu_\tau$ the measure generated by it, let
$E$ be a Borel subset of the interval $[a,b]$,
and write
$$
E_\infty = \{x \in E: \overline{D} \tau(x) = \infty \}
.
$$
Then $E_\infty $ is a Borel subset of $E$ and $\mu_\tau$
is $\sigma$--finite on $E \setminus E_\infty$.
If, moreover, $\mu_\tau$
is $\sigma$--finite on $E$ then $E_\infty$
has Lebesgue measure zero.
}
%%\end{thm}
\smallskip
\noindent {\bf Proof.}
The set $E \setminus E_\infty$ is the union of the
sequence of sets
$$
E_n = \{x \in E: \overline{D} \tau(x) < n \}
$$
As easy estimate shows that $\mu_\tau(E_n) \leq n(b-a)$
and this shows that $\mu_\tau$
is $\sigma$--finite on $E \setminus E_\infty$.
Standard arguments suffice to show that all sets here
are Borel (cf.~\cite[\S4.2]{thomson1}).
Suppose now that
$K$ is
any closed
subset of $E_\infty$
for which $\mu_\tau(K)< \infty$
and let $c>0$.
We may
choose a gauge $\delta$
on $K$ so that
$$
V(\tau,K,\delta) \leq \mu_\tau(K) + 1
.$$
The collection
$
{\cal C}
$
of all open subintervals $I$
of $(a,b)$
with the property that
$\tau(I)> c|I|$
and, for some
$x \in I$, $|I|<\delta(x)$
is a Vitali cover of $K$. Thus there must exist a disjoint
sequence $\{I_i\} \subset {\cal C}$
so that $$
|K| \leq \sum_{i}|I_i|
$$
(here $|K|$ denotes the Lebesgue measure of the set $K$).
This gives us that
$$
c|K|
\leq
\sum_{i}c |I_i|
\leq \sum_{i} \tau(I_i)
\leq
V(\tau,K,\delta) \leq \mu_\tau(K) + 1
.
$$
The inequality
$$
c|K|
\leq
\mu_\tau(K) + 1
$$
can be valid for all $c>0$ only if $|K|=0$. Thus there can be
no closed subsets of $E_\infty$ of positive Lebesgue measure
that have finite $\mu_\tau$ measure.
This proves the second assertion of the lemma.
\nobreak \hfill
$\blacksquare$
\smallskip
Using this lemma and Theorem~1 we can prove the following theorem
asserting a very weak condition under which a.e.~finiteness of
the derivate $\overline{D} \tau(x)$
can be concluded. Prior to the conjecture of Pfeffer this condition
might have seemed impossibly weak.
Higher dimensional variants of this theorem
may be found in
B.~Bongiorno et al.~\cite{bened3}
and
Buczolich and Pfeffer~\cite{ZoliWash3}.
\bigskip
\noindent
{\bf THEOREM~2.}
{\em
Let $\mu_\tau$ be a measure constructed
from
an interval function $\tau$ and let $E \subset [a,b]$
be Lebesgue measurable. If $\mu_\tau$ is $\sigma$--finite on all
closed subsets of $E$ that have zero Lebesgue measure, then
the set
$$
E_\infty = \{x \in E: \overline{D} \tau(x) = \infty \}
$$
has Lebesgue measure zero.
}
\smallskip
\noindent {\bf Proof.}
The set $E_\infty$ is measurable if $E$ is and so, to prove that
it has Lebesgue measure zero, it is enough to show that every
closed subset has Lebesgue measure zero.
This follows immediately from Theorem~1 and the Lemma.
\nobreak \hfill
$\blacksquare$
\begin{thebibliography}{11}
\bibitem{bened}
B.~Bongiorno,
W.~F.~Pfeffer,
and B.~S. Thomson,
A full descriptive definition of the gage integral,
Canadian Math.~Bull., 39(4) (1996), 390--401.
\bibitem{bened2}
B.~Bongiorno, L.~Di~Piazza and
V.~Skvortsov,
The essential variation of a function
and some convergence theorems,
Anal. Math. (1) 22 (1996), 3--12.
\bibitem{bened2a}
B.~Bongiorno, L.~Di~Piazza and
V.~Skvortsov,
A new full descriptive characterization of the
Denjoy-Perron integral,
Real Anal. Exch., 21 (1995/96), No.~2, 656--663.
\bibitem{bened3}
B.~Bongiorno, L.~Di~Piazza and
D.~Preiss,
Infinite variation and derivatives
in $\Reals^n$,
Journal of Math.~Anal.~Appl.,
224 (1998) no.~1, 22--33.
\bibitem{Bruckner}
A.~M.~Bruckner, {\it Differentiation of Real Functions},
Springer-Verlag (1978).
\bibitem{BBT} A.~M. Bruckner, J.~B. Bruckner and B.~S. Thomson,
{\it Real Analysis} Prentice-Hall (1996).
\bibitem{ZoliWash2} Z.~Buczolich and W.~F.~Pfeffer,
When absolutely continuous implies $\sigma$-finite,
Bull. Csi., Acad. Royale
Belgique, serie 6, 1-6 (1997), 155--160.
\bibitem{ZoliWash} Z.~Buczolich and W.~F.~Pfeffer,
Variations of additive functions,
Czech. Math.~J., 47 (122) (1997), no.~3, 525--555.
\bibitem{ZoliWash3} Z.~Buczolich and W.~F.~Pfeffer,
On absolute continuity,
Journal of Math.~Anal.~Appl.,
222, (1998) no.~1, 64--78.
\bibitem{edgar}G.~A.~Edgar,
{\em Measure, Topology, and Fractal Geometry,}
Springer-Verlag, New York (1990).
\bibitem{edgar2}G.~A.~Edgar,
{\em Integral, Probability, and Fractal Measures,}
Springer-Verlag, New York (1998).
\bibitem{Ene1}
V.~Ene, \emph{Real functions - current topics}, Lect.
Notes in Math., vol.
1603, Springer-Verlag, 1995.
\bibitem{Ene39}
V.~Ene, {Characterizations of {$VB^*G \cap
(N)$}}, Real Analysis
Exchange 23 (1997/87) no.~2, 571--599.
\bibitem{ene1}
V.~Ene, Thomson's variational measure,
Real Anal. Exch., (to appear).
\bibitem{Pfeffer}
W.~F.~Pfeffer, {\em The Riemann Approach to Integration:
Local Geometric Theory.}
Cambridge University Press (1993).
\bibitem{Pfefferi2}
W.~F.~Pfeffer, The generalized Riemann-Stieltjes integral,
Real Anal. Exch., Vol. 21, No.~2, 521-547.
\bibitem{Pfeffer3}
W.~F.~Pfeffer, On variations of functions of one real variable,
Comment.~Math.,Univ.~Carolin., 38,
no.~1(1997), 61-71.
\bibitem{Pfeffer4}
W.~F.~Pfeffer and B.S. Thomson, Measures defined by gages,
Canad. Journal of Math. 44 (6) 1992, 1303--1316.
\bibitem{Saks}
S.~Saks, {\em Theory of the Integral},
Dover, (1937).
\bibitem{thomson1}
B.~S.~Thomson,
{\em Derivates of Interval Functions,}
Memoir American Math. Soc., 452, Providence, 1991.
\bibitem{thomson2}
B.~S.~Thomson,
$\sigma$--finite Borel measures on the real line.
Real Anal. Exch., Vol. 23, (1997-98) no.~1, 185--192.
\end{thebibliography}
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\begin{document}
{\huge R}{\Large EAL} {\huge A}{\Large NALYSIS} {\huge E}{\Large
XCHANGE} \hfill {\sc St.~Olaf College}
\vskip -22pt
\rule{\textwidth}{.2mm}
\mbox{ } \hfill Northfield, Minnesota 55057
\thispagestyle{empty}
\hfill\today
Professor
Brian S. Thomson\newline Mathematics Department \newline Simon Fraser University\newline
Burnaby, B.~C.\newline Canada V5A~1S6.
Dear Professor Brian S. Thomson:
\noindent Beginning with Volume 23, the {\it Real Analysis Exchange\/} will be
printed by the Michigan State University Press. They have pointed out to us
that for
our protection and for yours, copyright ownership should be established.
That is the
purpose of this document.
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author and Michigan State University Press to publish the article
\begin{center}
%
{SOME PROPERTIES OF VARIATIONAL MEASURES}
%
\end{center}
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\hspace{2in} Signature\hspace{2.7in} Date
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{Some Properties of Variational Measures}
{Brian S. Thomson}
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{\huge R}{\Large EAL} {\huge A}{\Large NALYSIS} {\huge E}{\Large
XCHANGE} \hfill {\sc St.~Olaf College}
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Professor
Brian S. Thomson\newline Mathematics Department\newline Simon Fraser University\newline
Burnaby, B.~C.\newline Canada V5A~1S6.
Dear Professor Thomson:
Enclosed please find the ``gallies'' for
{\it SOME PROPERTIES OF VARIATIONAL MEASURES}
The title and authors as they will appear on the cover are:
\begin{itemize}
\item Cover Title:
\begin{center}
{Some Properties of Variational Measures}
\end{center}
\item Cover Author:
\begin{center}
{Brian S. Thomson}
\end{center}
\end{itemize}
Please make corrections in red on the enclosed copy and return it to
me within about two weeks, {\bf but no later than December 21.}.
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We prefer an e-mail response ({\bf in addition to the return of the
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The {\it Real Analysis Exchange} charges \$15 per page to those whose research
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Sincerely,
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Paul D. Humke, Editor \\
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