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특히, 본 논문은 Christensen가 abelian Polish group에서 Haar zero set을 정의하였으며, 이 정의는 separable 경우 shy set과 동치라고 주장합니다. 저자들은 prevalence concept를 사용하여 Lipschitz function의 differentiability를 연구했으며, Christensen의 main application으로 유한 차원 공간에 대한 Rademacher theorem의 analogue를 증명했습니다.
한편, 본 논문은 다른 학자들의 연구를 소개하며, 이들 연구가 거의 모든 곳에서 continuity와 differentiability가 성립한다는 것을 보여주었습니다. 또한, 저자들은 Tsujii가 manifold간의 함수 공간에 대해 measure zero를 정의하였으며, 이를 transversality 및 dynamical system에 관한 연구에 응용하였다고 주장합니다.
한글 요약 끝.
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arXiv:math/9304213v1 [math.FA] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 306-307PREVALENCE: AN ADDENDUMBRIAN R. HUNT, TIM SAUER, AND JAMES A. YORKESince the publication of our paper “Prevalence: a translation-invariant ‘almostevery’ on infinite-dimensional spaces” in this journal [7], we have become aware ofsome work that is closely related to ours. We wish to call the reader’s attention tothis material.
We thank J. Borwein, N. Kalton, and R. Dudley for informing us ofthis related work.We defined the notions of “prevalent” and “shy” to be used in infinite-dimensionalspaces of functions as analogues of the notions of “almost every” and “measure zero”with respect to Lebesgue measure on Euclidean spaces. Our definitions were givenfor complete metric linear spaces, and they extend trivially to abelian groups thatare not vector spaces (but still have the topology of a complete metric).
Thesedefinitions have been extended further to nonabelian groups by Mycielski [11].For “abelian Polish groups” (topological abelian groups with a complete separa-ble metric), Christensen [3] defined the notion of a “Haar zero set”. Christensen’sdefinition is equivalent to our definition of a shy set in the separable case; ourdefinition has an extra provision which is only relevant for nonseparable spaces.We believe there are many possible applications of these ideas.
Christensen’smain application [4, 5] is to prove an analogue of Rademacher’s theorem (that aLipschitz function from one Euclidean space to another is differentiable almost ev-erywhere) for Lipschitz functions on Banach spaces. (Although this result is nottrue for the Fr´echet derivative, it is for a slightly weaker notion of differentiability.
)For other results concerning almost everywhere differentiability of Lipschitz func-tions on Banach spaces, see [1, 2, 9, 10, 12, 13]; some of these papers offer differentnotions of “almost everywhere”. Christensen’s definition has also been used [6, 8] instudying the continuity and differentiability of convex functions on Banach spaces.In [7] we presented ten results involving prevalence, with emphasis on dynamicalsystems and related areas such as transversality.
The focus of our applications wasdifferent than that of the previous authors, who were primarily concerned with ar-bitrary Banach spaces. We were interested in proving that almost every function(or dynamical system) in a certain space (such as C1(Rn)) has a certain property.We made explicit the role that Lebesgue measure (on finite-dimensional subspaces)can play in proving such results, and we hope to have made accessible many moreresults of this type.Finally, we would like to mention that Tsujii [14] has formulated a definition of“measure zero” for spaces of functions from one manifold to another.
Tsujii givesseveral applications to transversality and dynamical systems [14–16].Received by the editors November 25, 1992.1991 Mathematics Subject Classification. Primary 28C20, 60B11; Secondary 58F14.c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1
2BRIAN R. HUNT, TIM SAUER, AND JAMES A. YORKEReferences1. N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math.57 (1976), 147–190.2.
J. M. Borwein, Minimal cuscos and subgradients of Lipschitz functions, Fixed Point Theoryand its Applications, (J.-B. Baillon and M. Thera, eds.
), Pitman Lecture Notes in Math., vol.252, Longman, Essex, 1991.3. J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math.13 (1972), 255–260.4., Measure theoretic zero sets in infinite dimensional spaces and applications to differ-entiability of Lipschitz mappings, Publ.
D´ep. Math.
(Lyon) 10 (1973), no. 2, 29–39.5., Topology and Borel structure, North-Holland, Amsterdam, 1974.6.
P. Fischer and Z. Slodkowski, Christensen zero sets and measurable convex functions, Proc.Amer. Math.
Soc. 79 (1980), 449–453.7.
B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: a translation-invariant “almost every”on infinite-dimensional spaces, Bull.
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Soc. 27 (1992), 217–238.8.
M. Jouak and L. Thibault, Directional derivatives and almost everywhere differentiability ofbiconvex and concave-convex operators, Math. Scand.
57 (1985), 215–224.9. P. Mankiewicz, On the differentiability of Lipschitz mappings in Fr´echet spaces, Studia Math.45 (1973), 15–29.10., On topological, Lipschitz, and uniform classification of LF-spaces, Studia Math.
52(1974), 109–142.11. J. Mycielski, Unsolved problems on the prevalence of ergodicity, instability and algebraicindependence, Ulam Quarterly (to appear).112.
R. R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces,Pacific J. Math.
77 (1978), 523–531.13. L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued func-tions, Nonlinear Anal.
Theory Methods Appl. 6 (1982), 1037–1053.14.
M. Tsujii, A measure on the space of smooth mappings and dynamical system theory, J. Math.Soc. Japan 44 (1992), 415–425.15., Rotation number and one-parameter families of circle diffeomorphisms, Ergodic The-ory Dynamical Systems 12 (1992), 359–363.16., Weak regularity of Lyapunov exponents in one dimensional dynamics, preprint.(B.
R. Hunt and J. A. Yorke) Institute for Physical Science and Technology, Uni-versity of Maryland, College Park, Maryland 20742E-mail address, B.
Hunt: hunt@ipst.umd.eduE-mail address , J. Yorke:yorke@ipst.umd.edu(T. Sauer) Department of Mathematics, George Mason University, Fairfax, Virginia22030E-mail address: tsauer@gmu.edu1The Ulam Quarterly is an electronic journal and is available in AMS-TEX and PostScript for-mat by anonymous ftp from math.ufl.edu in the directory pub/ulam. For information on orderingprinted copies, contact: Professor Piotr Blass, Editor-in-Chief, Ulam Quarterly, P. O.
Box 24708,W. Palm Beach, FL 33416-4708.
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