Transfinite exhaustion
Martin
One exercise from Halmos's Measure theory follows. My friend succeeded
to solve it by other means - so I'm not asking for the solution - but I
wonder what he means under transfinite exhaustion, so I wanted put here
the whole context of this expression.
Any reference to a book, site or some proof, which uses this method
would help.
Thanks in advance!
Martin
Halmos: Measure theory, p.174, exercise 2
If (S,mu) is a sigma-finite, non atomic measure ring, and if E0 in S,
then, for every extended real number a with 0 <= a <= mu(E0) there
exists an element E is S such that E subset E0 and mu(E)=a.
(Hint: Since the case a=infty is trivial, there is no loss of
generality in assuming that mu(E0)<infty. The desired result follows by
a transfinite exhaustion process. The method is similar to the one used
in proving that any two points in a complete, convex metric space may
be joined by a segment, and in fact the present assertion is a special
case of this general theorem in metric geometry.)
[If I'm not mistaken, the theorem Halmos mention's is known as Menger's
theorem.]
Stephen J. Herschkorn
> One exercise from Halmos's Measure theory follows. [...] I
> wonder what he means under transfinite exhaustion, so I wanted put here
> the whole context of this expression.
>
>
> If (S,mu) is a sigma-finite, non atomic measure ring, and if E0 in S,
> then, for every extended real number a with 0 <= a <= mu(E0) there
> exists an element E is S such that E subset E0 and mu(E)=a.
> (Hint: Since the case a=infty is trivial, there is no loss of
> generality in assuming that mu(E0)<infty. The desired result follows by
> a transfinite exhaustion process. The method is similar to the one used
> in proving that any two points in a complete, convex metric space may
> be joined by a segment, and in fact the present assertion is a special
> case of this general theorem in metric geometry.)
>
> [If I'm not mistaken, the theorem Halmos mention's is known as Menger's
> theorem.]
My guess is that transfinite exhaustion refers to Zorn's lemma. Halmos
often uses such phraseology in reference to induction through ordinals
or its equivalent (e.g., ZL). Does the proof you know use ZL?
-S.J. Herschkorn
Tutor on the Internet and in Central New Jersey and Manhattan
Martin
As far as I remember, he started using the simple fact that between 2
values we can find another one and then he was able to approximate
every value somehow... or something like that.
My interest in this term is that I recently made some proofs using this
method - I construct some system by transfinite induction and at some
step the transfinite process stops - and at this step I obtain the
object, whose existence I wanted to prove. The argument, why I must
stop at some point, is that otherwise there would be a proper class of
some objects - which is a contradiction.
It seems quite possible to rewrite proofs like these using ZL (actually
I'm too lazy to write it down).
So I hoped to find analogous kind of proof somewhere - to persuade
myself, that this method is kind of standard.
Martin
Dave L. Renfro
If you look in the "References" section at the back of Halmos'
book, specifically on p. 291 for Section 41 (the section that
the exercise you cited appears in), Halmos cites pp. 85-87
of his reference [50]. This reference (Halmos, p. 295) is a
1928 paper by K. Menger in Mathematische Annalen (volume 100),
a journal that most every university library will have. However,
the older volumes are also publicly available on the internet at
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D25917.html
Volume 100 is at
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D29345.html
and Menger's paper begins on p. 75.
Since I can't read German, my personal preference would be
to go to Blumenthal's book "Theory and Applications of
Distance Geometry". In Chapter II "Metric Segments and Lines",
Section 14 "Convexity in metric spaces. Metric segments",
Theorem 14.1 (p. 41) says: "Each two distinct points of a
complete and convex metric space M are joined by a metric
segment of the space." The paragraph before this theorem is:
"It was proved by Menger that each two points of a
closed and convex subset of a compact metric space
are joined by a metric segment belonging to the
subset. Using transfinite induction, he established
this result also for complete metric spaces. The
proof we shall give for this fundamental theorem,
which avoids transfinite induction, is due to
Aronszajn."
Incidentally, there are a number of results that were first
proved by transfinite induction and then later, other proofs
were found (the proofs that make it into monographs and
textbooks). Many well known theorems in functional analysis
are examples of this, as well as Borel's original proof of
the Heine-Borel theorem in 1895 (the version that says any
countable covering of [a,b] by open intervals has a finite
subcover of [a,b]). (Borel's original proof appears at the
end of his 1894, published in 1895, Ph.D. Thesis, but by the
time Borel wrote his influencial 1898 text "Lecons sur la
Fonctions de Variables Réelles", he had come up with a proof
that avoided, at least explicitly, transfinite induction.)
The result in Halmos is sometimes called the Darboux property,
or the intermediate value property, for measures. For other
proofs of this result, see:
[1] Nicolae Dinculeanu, VECTOR MEASURES, Pergamon Press, 1967.
[See p. 26.]
[2] Paul R. Halmos, "On the set of values of a finite measure",
Bulletin of the American Mathematical Society 53 (1947),
138-141.
[3] Hans Hahn and Arthur Rosenthal, SET FUNCTIONS, The University
of New Mexico Press, 1948. [See Section I.6, pp. 51-53.]
[4] Jaroslav Lukes and Jan Maly, MEASURE AND INTEGRAL, MATFYZPRESS
(Charles University, Czech Republic), 1995.
[See p. 9, Exercise 2.15.]
[5] Alan J. Weir, GENERAL INTEGRATION AND MEASURE, Cambridge
University Press, 1974.
[See Problem 19, p. 100; solution on p. 264.]
For related results, see:
[6] Julius Rubin Blum, "On relatively nonatomic measures",
Proceedings of the American Mathematical Society 12 (1961),
457-459.
[7] Ronald John Nunke and Leonard Jimmie Savage, "On the set of
values of a nonatomic finitely additive, finite measure",
Proceedings of the American Mathematical Society 3 (1952),
217-218.
Nunke/Savage prove that if (X,A,mu) is any _finitely_ additive
measure space with no atoms, then there exists a finitely
additive measure mu' on (X,A) such that m'(X) = 4 and no
elements of the algebra A have m'-measure belonging to the
open interval (1,3).
A higher-dimensional generalization of this result was proved
in 1940 by A. A. Liapounoff. A couple of references are:
[8] Andrew M. Bruckner, Judith B. Bruckner, and Brian S. Thomson,
REAL ANALYSIS, Prentice-Hall, 1997. [See Chapter 2.13, p. 118,
Exercise 2:13.8 "Liaponoff's (sic) theorem"]
[9] R. E. Jamison, "A quick proof for a one-dimensional version
of Liapounoff's theorem", American Mathematical Monthly
81 (1974), 507-508.
Many more references for Liapounoff's theorem are given
in the first URL below. The next two URL's include papers
about Liapounoff's theorem, but these two lists will include
other things as well. (Liapounoff's name is associated with
other things than his convexity result for the range of
a sequence of measures.)
http://www.math.gatech.edu/~hill/publications/reflist.html
http://www.emis.de/projects/EULER/search?q=ti%3ALiapounoff
http://www.emis.de/projects/EULER/search?q=ti%3ALiapounov
http://scholar.google.com/scholar?q=Liapounoff+measure+convex
http://scholar.google.com/scholar?q=Liapounov+measure+convex
http://books.google.com/books?q=Liapounoff+measure+convex
http://books.google.com/books?q=Liapounov+measure+convex
Dave L. Renfro
Martin
I've downloaded the Menger's paper and I'll have a look at the
mentioned proof.
Martin