Abstract
For a number ɛ > 0 and a real function f on an interval [a, b], denote by N(ɛ, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a i , b i ], i = 1, …, n, on [a, b] such that |f(a i ) − f(b i )| > ɛ for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(ɛ) ≡ lim supj→∞ N(ɛ, f j , [a, b]) < ∞ for any ɛ > 0, then the sequence {f j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(ɛ, f, [a, b]) ≤ n(ɛ) for any ɛ > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.
Similar content being viewed by others
References
I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974) [in Russian].
R. M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and p-Variation, With the collaboration of Jinghua Qian, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1999), Vol. 1703
S. Gniłka, “On the generalized Helly’s theorem,” Funct. Approximatio Comment. Math. 4, 109–112 (1976).
J. Musielak and W. Orlicz, “On generalized variations. I,” Studia Math. 18, 11–41 (1959).
V. V. Chistyakov, “On multivalued mappings of finite generalized variation,” Mat. Zametki 71(4), 611–632 (2002) [Math. Notes 71 (3–4), 556–575 (2002)].
V. V. Chistyakov, “Selections of bounded variation,” J. Appl. Anal. 10(1), 1–82 (2004).
V. V. Chistyakov, “A selection principle for functions of a real variable,” Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 53(1), 25–43 (2005).
M. Schramm, “Functions of ϕ-bounded variation and Riemann-Stieltjes integration,” Trans. Amer. Math. Soc. 287(1), 49–63 (1985).
D. Waterman, “On Λ-bounded variation,” Studia Math. 57(1), 33–45 (1976).
V. V. Chistyakov, “The optimal form of selection principles for functions of a real variable,” J. Math. Anal. Appl. 310(2), 609–625 (2005).
V. V. Chistyakov, “A selection principle for functions with values in a uniform space,” Dokl. Ross. Akad. Nauk 409(5), 591–593 (2006) [Doklady Math. 74 (1), 559–561 (2006)].
V. V. Chistyakov, “A pointwise selection principle for functions of a single variable with values in a uniform space,” Mat. Tr. 9(1), 176–204 (2006) [Siberian Adv. Math. 16 (3), 15–41 (2006)].
Z. A. Chanturiya [Čanturija], “Themoduli of variation of a function and its applications in the theory of Fourier series,” Dokl. Akad. Nauk SSSR 214(1), 63–66 (1974).
K. Schrader, “A generalization of the Helly selection theorem,” Bull. Amer. Math. Soc. 78(3), 415–419 (1972).
L. Di Piazza and C. Maniscalco, “Selection theorems, based on generalized variation and oscillation,” Rend. Circ. Mat. Palermo (2) 35(3), 386–396 (1986).
C. Maniscalco, “A comparison of three recent selection theorems,” Math. Bohem. 132(2), 177–183 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu. V. Tret’yachenko, V. V. Chistyakov, 2008, published in Matematicheskie Zametki, 2008, Vol. 84, No. 3, pp. 428–439.
Rights and permissions
About this article
Cite this article
Tret’yachenko, Y.V., Chistyakov, V.V. Selection principle for pointwise bounded sequences of functions. Math Notes 84, 396–406 (2008). https://doi.org/10.1134/S0001434608090101
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434608090101