"measure zero" on infinite dimensional spaces
Alexander R Pruss
I remember reading a number of years, I think in the AMS Bulletin,
about work on defining a notion of "measure zero" on infinite dimensional
spaces. Of course, a measure that has all the requisite properties doesn't
exist, but a notion of negligibility can still be defined and give
intuitively obvious results like "C[0,1] has 'measure zero' in L^p[0,1]".
Does someone know what the latest work in this area is? A reference to a
survey piece would be greatly appreciated.
Thanks!
Alex
--
Dr. Alexander R. Pruss || e-mail: pru...@pitt.edu
Graduate Student || home page: http://www.pitt.edu/~pruss
Department of Philosophy || alternate e-mail address: pr...@member.ams.org
University of Pittsburgh || Erdos number: 4
Pittsburgh, PA 15260 ||
U.S.A. ||
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"Philosophiam discimus non ut tantum sciamus, sed ut boni efficiamur."
- Paul of Worczyn (1424)
Randall Dougherty
Alexander R Pruss <pru...@pitt.edu> wrote:
> I remember reading a number of years, I think in the AMS Bulletin,
>about work on defining a notion of "measure zero" on infinite dimensional
>spaces. Of course, a measure that has all the requisite properties doesn't
>exist, but a notion of negligibility can still be defined and give
>intuitively obvious results like "C[0,1] has 'measure zero' in L^p[0,1]".
>Does someone know what the latest work in this area is? A reference to a
>survey piece would be greatly appreciated.
The survey paper you're recalling is:
B. Hunt, T. Sauer, and J. Yorke, "Prevalence: a translation-invariant
'almost every' on infinite-dimensional spaces", Bull. Amer. Math. Soc. 27
(1992), 217-238; addendum, Bull. Amer. Math. Soc. 28 (1993), 306-307.
Here are some other relevant papers:
J. P. R. Christensen, "On sets of Haar measure zero in abelian Polish
groups", Israel J. Math. 13 (1972), 255-260.
R. Dougherty, "Examples of non-shy sets", Fund. Math. 144 (1994), 73-88.
B. Hunt, "The prevalence of continuous nowhere differentiable functions",
Proc. Amer. Math. Soc. 122 (1994), 711-717.
J. Mycielski, "Some unsolved problems on the prevalence of ergodicity,
instability and algebraic independence", Ulam Quarterly 1 (1992) no. 3,
30-37.
S. Solecki, "On Haar null sets", Fund. Math. 149 (1996), 205-210.
Randall Dougherty r...@math.ohio-state.edu
Department of Mathematics, Ohio State University, Columbus, OH 43210 USA
"I have yet to see any problem, however complicated, that when looked at in the
right way didn't become still more complicated." Poul Anderson, "Call Me Joe"
G. A. Edgar
> Hi!
>
> I remember reading a number of years, I think in the AMS Bulletin,
> about work on defining a notion of "measure zero" on infinite dimensional
> spaces. Of course, a measure that has all the requisite properties doesn't
> exist, but a notion of negligibility can still be defined and give
> intuitively obvious results like "C[0,1] has 'measure zero' in L^p[0,1]".
> Does someone know what the latest work in this area is? A reference to a
> survey piece would be greatly appreciated.
>
>
That Bulletin article duplicated unknowingly Christensen's earlier
work on so-called "Haar zero sets"...
Christensen, Israel J. Math. 13 (1972) 255--260.
Hunt, Sauer, Yorke, Bull. Amer. Math. Soc. 27 (1992) 217--238;
Bull. Amer. Math. Soc. 28 (1993) 306--307.
--
Gerald A. Edgar ed...@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)
Mark C McClure
: about work on defining a notion of "measure zero" on infinite dimensional
: spaces.
You are probably referring to
Brian Hunt, Tim Sauer, and James Yorke. Prevalence: A Translation
Invariant ``Almost Every'' on Infinite Dimensional Spaces, Bulletin
of the American Mathematical Society, 27 (1992), 217 - 38.
It turns out that their concept is equivalent to a much older concept.
See:
J.~P.~R.~Christensen, ``On Sets of Haar Measure
Zero in Abelian Polish Groups'', Israel J.~Math. {\bf 13} (1972),
255-60.
Hunt, Sauer, and Yorke have a somewhat different objective however.
There are a number of theorems of the form "The typical continuous
functions satifies property X". Typicality being measured by Baire
category. Prevalence is a measure theoretic property. Thus they
state a number of theorems of the form "The prevalent continuous
functions satifies property X". Here are a couple of other papers
following this scheme:
Hunt, Brian R. The prevalence of continuous nowhere differentiable
functions. Proc. Amer. Math. Soc. 122 (1994), no. 3, 711--717
McClure, Mark The prevalent dimension of graphs. Real Anal.
Exchange 23 (1997/98), no. 1, 241--246.
Hope that helps,
--
__/\__
Mark McClure \ /
Department of Mathematics __/\__/ \__/\__
UNC - Asheville \ /
Asheville, NC 28804 /__ __\
http://www.unca.edu/~mcmcclur/ \ /
__/\__ __/ \__ __/\__
\ / \ / \ /
__/\__/ \__/\__/ \__/\__/ \__/\__
Dave L. Renfro
<http://forum.swarthmore.edu/epigone/sci.math.research/cleekhenbrum>
wrote
>Hi!
>
> I remember reading a number of years, I think in the AMS
>Bulletin, about work on defining a notion of "measure zero" on
>infinite dimensional spaces. Of course, a measure that has all
>the requisite properties doesn't exist, but a notion of negligibility
>can still be defined and give intuitively obvious results like
>"C[0,1] has 'measure zero' in L^p[0,1]". Does someone know what
>the latest work in this area is? A reference to a survey piece
>would be greatly appreciated.
>
>Thanks!
>Alex
Some others (Dougherty, McClure, Edgar) have already
given a few older references, so I'll focus on the more
recent and/or lesser known items that might be of interest.
In R^n the notion "Haar null" is equivalent to having
Lebesgue n-measure zero. Thus, the notion is essentially
independent of the Baire category notion of smallness.
[I say *essentially* because, for instance, every F_sigma
measure zero set is a first category set.] Indeed, the
two notions are orthogonal in the sense that R^n can be
expressed as a union of a first category set and a measure
zero set.
This decomposition continues to hold in separable Banach spaces.
Preiss/Tiser (theorem 1 in [10]) prove that if X is an infinite
dimensional separable Banach space, then we can write
X = A union B, where A is a countable union of closed porous
sets and B has linear measure zero on every line.
In particular, B is a Haar null set, B is negligible
in the sense of Aronszajn, and B has zero measure for any
non-degenerate Gaussian measure on X.
[The condition "is a countable union of closed porous sets"
is strictly stronger than "is a countable union of porous
sets", which in turn is strictly stronger than "first category".]
Incidentally, Csornyei [4] has proved that the sigma-ideals of
Aronszajn null sets and Gaussian null sets coincide in any
separable Banach space.
Hongjian Shi's 1997 Ph.D. Dissertation [12] gives the most
comprehensive survey of Haar null notions that I am aware of.
I've included a summary of some of the results from Shi's
Dissertation at the end of this post. [These remarks are
taken from the annotated bibliography of a book manuscript
on porous sets that I've been working on.] {Yes, I'm aware
of the coincidence with "shy" and "Shi", and so is Shi!}
Shawn Wang's 1999 Ph.D. Dissertation [13] includes a few
results similar to those in Shi's Dissertation, but the
focus here is primarily on other matters rather than on
Haar null results.
Finally, Jan Kolar's Ph.D. Dissertation [7] deals with a
simultaneous strengthening of Haar null and sigma-porosity.
(I believe he has proved his notion is strictly
stronger than their conjunction.) Using this notion, he
has proved several rather strong nowhere differentiability
results (whose statements involve various disagreements of
the four Dini derivates at each point on sets having
unilateral upper Lebesgue density 1/2) for
"super-almost all" (a term I just now made up)
continuous functions f : [0,1] --> R (sup norm).
*************************************************************
*************************************************************
1. Jonathan M. Borwein and S. P. Fitzpatrick, "Closed convex Haar
null sets", CECM Preprint 95:052 at
<http://www.cecm.sfu.ca/preprints/1995pp.html>.
[If E is a separable super-reflexive Banach space then every
closed convex subset of E with empty interior is a Haar null set.]
2. Jonathan M. Borwein and Warren B. Moors, "Null sets and
essentially smooth Lipschitz functions", SIAM J. Optim. 8
(1998), 309-323.
CECM Preprint 96:068 at
<http://www.cecm.sfu.ca/preprints/1996pp.html>
3. Janusz Brzdek, "The Christensen measurable solutions of a
generalization of the Golab-Schinzel functional equation", Ann.
Polonici Math. 64 (1996), 195-205.
4. Marianna Csornyei, "Aronszajn null and Gaussian null sets
coincide",
to appear in Israel J. Math. [This has probably already appeared.
(It's been several months since I've visited a research library.)]
5. M. Grinc, "On measure zero sets in topological vector spaces",
Acta Math. Univ. Commenianae 65 (1996), 87-91.
6. V. Yu. Kaloshin, "Prevalence in the space of finitely smooth
maps", Functional Analysis and its Applications 31(2) (1997),
95-99.
7. Jan Kolar, Ph.D. Dissertation (under Ludek Zajicek), Charles
University, Czech Republic. [Recently completed, or soon to
be completed.]
8. Eva Matouskova, "Convexity and Haar null sets", Proc. Amer. Math.
Soc. 125 (1997), 1793-1799.
9. Eva Matouskova, "The Banach-Saks property and Haar null sets",
Comment. Math. Univ. Carolinae 39 (1998), 71-80. [Abstract: A
characterization of Haar null sets in the sense of Christensen is
given. Using it, we show that if the dual of a Banach space X has
the Banach-Saks property, then closed and convex subsets of X with
empty interior are Haar null.]
10. David Preiss and Jaroslav Tiser, "Two unexpected examples
concerning differentiability of Lipschitz functions on Banach
spaces", pp. 219-238 in Geometric Aspects of Functional Analysis,
ed. by J. Lindenstrauss and V. Milman, Operator Theory: Advances
and Applications 77, Birkhauser Verlag, 1995.
11. Timothy D. Sauer and James A. Yorke, "Are the dimensions of a
set and its image equal under typical smooth functions?",
Ergodic Theory and Dynamical Systems 17 (1997), 941-956.
12. Hongjian Shi, "Measure-Theoretic Notions of Prevalence",
Ph.D. Dissertation (under Brian S. Thomson), Simon Fraser
University, October 1997, ix + 165 pages.
13. Shawn X. Wang, "Fine and Pathological Properties of
Subdifferentials", Ph.D. Dissertation (under Jonathan M. Borwein),
Simon Fraser University, August 1999, approx. 180 pages.
CECM Preprint 99:134 at
<http://www.cecm.sfu.ca/preprints/1999pp.html>
**************************************************************
**************************************************************
A SUMMARY OF HONGJIAN SHI'S PH.D. DISSERTATION
A triple (i.j.k) of positive integers refers to Shi's
"chapter-section-item" numbering of theorems.
The most thorough survey on variations (many of which are new)
of Haar null set notions that I am aware of. In addition, many
prevalent properties in various function spaces are established,
including the following. Let C[0,1] be the Banach space of
continuous functions f: [0,1] --> R with the supremum
norm and let D[0,1] be the Banach space of functions f: [0,1] --> R
with f(0) = 0 having a bounded derivative, where the norm of f
in D[0,1] is defined to be the supremum norm of the derivative
f'. (4.3.2) The set of continuous functions f of nowhere monotonic
type (i.e. for each c in R, f(x) - cx is not monotone on any
interval) is prevalent in C[0,1]. [REMARK: Brian Hunt proved
(Proc. AMS 122 (1994), 711-717) the stronger result that the set
of continuous functions f such that f does not satisfy a
*pointwise* Lipschitz condition at each point in [0,1] is
prevalent. However, as Shi points out, Hunt's proof shows the
exceptional set is shy using a two-dimensional probe, whereas
Shi is able to prove the set of continuous functions satisfying
the weaker property (being of nowhere monotonic type) is the
complement of a set that is shy using a one--dimensional probe.
It is an open problem (this is Shi's Problem 1, p. 13) whether
there exists a shy set in a separable Banach space that cannot
be proved shy using a 1-dimensional probe. Shi does prove
(3.2.4) that these notions are equivalent in R^2.]
(4.4.2) Let mu be a sigma-finite Borel measure on [0,1] and let
F be a linear subspace of the Banach space X of bounded
functions f: [0,1] --> R equipped with the supremum norm.
The set of functions in X that are discontinuous mu-almost
everywhere in [0,1] is either empty or prevalent in X.
[Hence, the prevalent function is discontinuous mu-almost
everywhere in the space of bounded approximately continuous
functions, in the space of bounded Darboux Baire 1 functions,
or in the space of bounded Baire 1 functions.]
(4.5.1) The prevalent (and the Baire-typical) function in
D[0,1] is monotone on some subinterval of [0,1].
(4.5.2) Let mu be a sigma-finite Borel measure on [0,1].
The derivative of the prevalent function in D[0,1] is
discontinuous mu-almost everywhere. (4.6.6) The prevalent
[(4.6.3) The Baire-typical] function in the space of Riemann
integrable functions (sup norm) has a c-dense set of
discontinuities. Chapter 5 gives an extensive treatment of
various prevalent notions in the space of homeomorphisms
h: [0,1] --> [0,1] such that h(0) = 0 and h(1) = 1 (sup norm).
In particular, several "natural" examples of the following
are found in this space: a meager set that is non-shy,
a non-shy set that is both left shy and right shy
(this space of homeomorphisms is a non-Abelian Polish group),
and a residual set that is not prevalent.
Dave L. Renfro <dlre...@gateway.net>
Department of Mathematics and Computer Science
Drake University
Des Moines, Iowa 50311
Dave L. Renfro
Haar null sets I overlooked the following paper:
Hongjian Shi and Brian S. Thomson, "Haar null sets in the
space of automorphisms on [0,1]", Real Analysis Exchange
24 (1998-99), 337-350.
The main goal of this paper seems to be to initiate
a study of Haar null sets in non-Abelian Polish groups
by studying them in one of the more important
examples of a non-Abelian Polish group. (See
Chapter 13 of John Oxtoby's book "Measure and
Category" for an introduction to the space of
automorphisms on [0,1] that is particularly
useful for the Shi/Thomson paper.)
Indeed, since every Polish group is isomorphic to a
closed subgroup of the automorphisms of the Hilbert
cube (see [1] or pp. 160-161 of [2]), the space
of automorphisms on [0,1] is a natural starting
point for such a study.
1. V. V. Uspenskii, "A universal topological
group with a countable base", Functional Analysis
and its Applications 20 (1986), 160-161.
2. Alexander Kechris, "Classical Descriptive Set
Theory", Springer-Verlag, 1995.