From: rld@math.ohio-state.edu (Randall Dougherty) Subject: Re: "measure zero" on infinite dimensional spaces Date: 13 Oct 1999 19:09:23 GMT Newsgroups: sci.math.research In article <7u2g1i$kms$2@usenet01.srv.cis.pitt.edu>, Alexander R Pruss wrote: > I remember reading a number of years, I think in the AMS Bulletin, >about work on defining a notion of "measure zero" on infinite dimensional >spaces. Of course, a measure that has all the requisite properties doesn't >exist, but a notion of negligibility can still be defined and give >intuitively obvious results like "C[0,1] has 'measure zero' in L^p[0,1]". >Does someone know what the latest work in this area is? A reference to a >survey piece would be greatly appreciated. The survey paper you're recalling is: B. Hunt, T. Sauer, and J. Yorke, "Prevalence: a translation-invariant 'almost every' on infinite-dimensional spaces", Bull. Amer. Math. Soc. 27 (1992), 217-238; addendum, Bull. Amer. Math. Soc. 28 (1993), 306-307. Here are some other relevant papers: J. P. R. Christensen, "On sets of Haar measure zero in abelian Polish groups", Israel J. Math. 13 (1972), 255-260. R. Dougherty, "Examples of non-shy sets", Fund. Math. 144 (1994), 73-88. B. Hunt, "The prevalence of continuous nowhere differentiable functions", Proc. Amer. Math. Soc. 122 (1994), 711-717. J. Mycielski, "Some unsolved problems on the prevalence of ergodicity, instability and algebraic independence", Ulam Quarterly 1 (1992) no. 3, 30-37. S. Solecki, "On Haar null sets", Fund. Math. 149 (1996), 205-210. Randall Dougherty rld@math.ohio-state.edu Department of Mathematics, Ohio State University, Columbus, OH 43210 USA "I have yet to see any problem, however complicated, that when looked at in the right way didn't become still more complicated." Poul Anderson, "Call Me Joe" ============================================================================== From: "G. A. Edgar" Subject: Re: "measure zero" on infinite dimensional spaces Date: Wed, 13 Oct 1999 15:43:44 -0400 Newsgroups: sci.math.research In article <7u2g1i$kms$2@usenet01.srv.cis.pitt.edu>, Alexander R Pruss 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. > > 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 edgar@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) ============================================================================== From: Mark C McClure Subject: Re: "measure zero" on infinite dimensional spaces Date: Wed, 13 Oct 1999 21:57:32 +0000 (GMT) Newsgroups: sci.math.research Alexander R Pruss 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. 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/ \ / __/\__ __/ \__ __/\__ \ / \ / \ / __/\__/ \__/\__/ \__/\__/ \__/\__ ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: "measure zero" on infinite dimensional spaces Date: 17 Oct 1999 14:05:13 -0400 Newsgroups: sci.math.research Alexander R Pruss [sci.math.research 13 Oct 1999 17:37:54 GMT] 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 . [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 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 ************************************************************** ************************************************************** 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 Department of Mathematics and Computer Science Drake University Des Moines, Iowa 50311 ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: "measure zero" on infinite dimensional spaces Date: 18 Oct 1999 10:23:45 -0400 Newsgroups: sci.math.research In my post surveying some recent work involving 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. Dave L. Renfro Department of Mathematics and Computer Science Drake University Des Moines, Iowa 50311