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Automatic continuity via analytic thinning

Authors: N. H. Bingham and A. J. Ostaszewski
Journal: Proc. Amer. Math. Soc. 138 (2010), 907-919
MSC (2000): Primary 26A03
Published electronically: November 5, 2009
MathSciNet review: 2566557
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Abstract: We use Choquet's analytic capacitability theorem and the Kestelman-Borwein-Ditor theorem (on the inclusion of null sequences by translation) to derive results on `analytic automaticity' - for instance, a stronger common generalization of the Jones/Kominek theorems that an additive function whose restriction is continuous/bounded on an analytic set $ T$ spanning $ \mathbb{R}$ (e.g., containing a Hamel basis) is continuous on $ \mathbb{R}$. We obtain results on `compact spannability' - the ability of compact sets to span $ \mathbb{R}$. From this, we derive Jones' Theorem from Kominek's. We cite several applications, including the Uniform Convergence Theorem of regular variation.

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  • [AD] J. Aczél and J. Dhombres, Functional equations in several variables, Encycl. Math. Appl. 31, Cambridge University Press, Cambridge, 1989. MR 1004465 (90h:39001)
  • [Ban] S. Banach, Ueber metrische Gruppen, Studia Math. III (1931), 101-113; reprinted in Oeuvres avec des commentaires, vol. II, pp. 401-411, PWN, Warsaw, 1979.MR 0563126 (81f:46001)
  • [BGT] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Encycl. Math. Appl., 27, Cambridge University Press, Cambridge, 1987. MR 898871 (88i:26004)
  • [BOst] N. H. Bingham and A. J. Ostaszewski, Generic subadditive functions, Proc. Amer. Math. Soc. 136 (2008), 4257-4266. MR 2431038 (2009e:39025)
  • [BOst3] N. H. Bingham and A. J. Ostaszewski, Infinite combinatorics and the theorems of Steinhaus and Ostrowski, CDAM Research Report Series, LSE-CDAM-2007-15, London School of Economics, 2007.
  • [BOst6] N. H. Bingham and A. J. Ostaszewski, New automatic properties: subadditivity, convexity, uniformity, Aequationes Math., to appear.
  • [BOst9] N. H. Bingham and A. J. Ostaszewski, Infinite combinatorics in function spaces: category methods, Publ. Inst. Math. Bé ograd, to appear.
  • [BOst11] N. H. Bingham and A. J. Ostaszewski, Bitopology and measure-category duality, CDAM Research Report Series, LSE-CDAM-2007-29, London School of Economics, 2007.
  • [BOst12] N. H. Bingham and A. J. Ostaszewski, Normed groups: dichotomy and duality, CDAM Research Report Series, LSE-CDAM-2008-10, London School of Economics, 2008.
  • [Bl] A. Blass, Existence of bases implies the axiom of choice, pp. 31-33 in J. E. Baumgartner, D. A. Martin and S. Shelah (eds.), Axiomatic set theory, Contemporary Mathematics, 31, Amer. Math. Soc., Providence, RI, 1984. MR 763890 (86a:04001)
  • [BoDi] D. Borwein and S. Z. Ditor, Translates of sequences in sets of positive measure, Canadian Mathematical Bulletin 21 (1978), 497-498. MR 523593 (80i:28018)
  • [Bou] N. Bourbaki, Elements of mathematics: General topology, Part 1, Addison-Wesley, Reading, MA, 1966. MR 0205210 (34:5044a)
  • [Ch1] G. Choquet, Capacités. Premières dé finitions, Comptes Rendus Acad. Sci. Paris 234 (1952), 35-37.MR 0045261 (13:555b)
  • [Ch2] -, Extension et restriction d'une capacité, Comptes Rendus Acad. Sci. Paris 234 (1952), 383-385. MR 0045784 (13:633d)
  • [Ch3] -, Propriétés fonctionnelles des capacités alternées ou monotones. Exemples. Comptes Rendus Acad. Sci. Paris 234 (1952), 498-500. MR 0047114 (13:829e)
  • [Ch4] -, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953-54), 131-295. MR 0080760 (18:295g)
  • [Dar] G. Darboux, Sur la composition des forces en statiques, Bull. des Sci. Math. 9 (1875), 281-288.
  • [Dav] R. O. Davies, Subsets of finite measure in analytic sets, Nederl. Akad. Wetensch. Proc. Ser. A. 55; Indagationes Math. 14 (1952), 488-489. MR 0053184 (14:733g)
  • [Del1] C. Dellacherie, Capacités et processus stochastiques, Ergeb. Math., Springer-Verlag, Berlin, Heidelberg, New York, 1972. MR56:6810 MR 0448504 (56:6810)
  • [Del2] C. Dellacherie, Un cours sur les ensembles analytiques, Part II (pp. 183-316 of [Rog2]).
  • [Eng] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • [EKR] P. Erdős, H. Kestelman and C. A. Rogers, An intersection property of sets with positive measure, Coll. Math. 11 (1963), 75-80. MR 0158961 (28:2182)
  • [Fal] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986. MR 867284 (88d:28001)
  • [Jones] F. B. Jones, Measure and other properties of a Hamel basis, Bull. Amer. Math. Soc. 48 (1942), 472-481. MR 0006555 (4:4e)
  • [Kah] J.-P. Kahane, Probabilities and Baire's theory in harmonic analysis, Twentieth century harmonic analysis--a celebration (Il Ciocco, 2000), pp. 57-72, NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer Acad. Publ., Dordrecht, 2001. MR 1858779 (2002g:42002)
  • [Kech] A. S. Kechris, Classical descriptive set theory , Graduate Texts in Mathematics, 156, Springer, New York, 1995. MR 1321597 (96e:03057)
  • [Kel] J. L. Kelley, General topology, Van Nostrand, New York, 1955. MR 0070144 (16:1136c)
  • [Kes] H. Kestelman, The convergent sequences belonging to a set, J. London Math. Soc.22 (1947), 130-136. MR 0022893 (9:274j)
  • [Kom1] Z. Kominek, On the sum and difference of two sets in topological vector spaces, Fund. Math. 71 (1971), no. 2, 165-169. MR 0293363 (45:2440)
  • [Kom2] Z. Kominek, On the continuity of Q-convex and additive functions, Aeq. Math. 23 (1981), 146-150. MR 689027 (84e:26012)
  • [Kucz] M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy's functional equation and Jensen's inequality, PWN, Warsaw, 1985. MR 788497 (86i:39008)
  • [Kur1] K. Kuratowski, Topology, Vol. I, Academic Press, New York-London; PWN, Warsaw, 1966. MR 0217751 (36:840)
  • [Kur2] K. Kuratowski, Topology, Vol. II, Academic Press, New York-London; PWN, Warsaw, 1968. MR 0259835 (41:4467)
  • [Lacz] M. Laczkovich, Analytic subgroups of the reals , Proc. Amer. Math. Soc. 126 (1998), 1783-1790. MR 1443837 (98g:04001)
  • [Ostr] A. Ostrowski, Mathematische Miszellen XIV: Ü ber die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichungen, Jahresb. Deutsch. Math. Ver. 38 (1929), 54-62 (reprinted in Collected papers of Alexander Ostrowski, Vol. 4, pp. 49-57, Birkhäuser, Basel, 1984).MR 0760994 (86m:01075d)
  • [Pet] Gy. Petruska, On Borel sets with small cover, Real Analysis Exchange 18 (1992-93), no. 2, 330-338. MR 1228398 (95g:28003a)
  • [Pet1] B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of Math. (2) 52 (1950), 293-308. MR 0038358 (12:391d)
  • [Pic1] S. Piccard, Sur les ensembles de distances des ensembles de points d'un espace Euclidien, Mém. Univ. Neuchâtel, 13, Secrétariat de l'Université, Neuchâtel, 1939. MR 0002901 (2:129d)
  • [Pic2] S. Piccard, Sur des ensembles parfaites, Mém. Univ. Neuchâtel, 16, Secrétariat de l'Université, Neuchâtel, 1942. MR 0008835 (5:61n)
  • [Rog1] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862 (43:7576)
  • [Rog2] C. A. Rogers, J. Jayne, C. Dellacherie, F. Topsøe, J. Hoffmann-Jørgensen, D. A. Martin, A. S. Kechris and A. H. Stone, Analytic sets, Academic Press, London, 1980.
  • [Sierp1] W. Sierpiński, Sur un problème de M. Lusin, Giornale di Mat. di Battaglini (3) 7 (1917), 272-277 (reprinted as [Sierp2], II, pp. 166-170).
  • [Sierp2] W. Sierpiński, Oeuvres choisis, Vol. I (1974), II (1975), III (1976), PWN, Warsaw. MR 0414302 (54:2405), MR 0414303 (54:2406), MR 0414304 (54:2407)
  • [Si] M. Sion, Topological and measure theoretic properties of analytic sets, Proc. Amer. Math. Soc. 11, no. 5 (1960), 769-776. MR 0131509 (24:A1359)
  • [Sol1] S. Solecki, Covering analytic sets by families of closed sets, Journal of Symbolic Logic 59 (1994), 1022-1031. MR 1295987 (95g:54033)
  • [Sol2] S. Solecki, Analytic ideals and their applications, Ann. Pure and Applied Logic 99 (1999), 51-72. MR 1708146 (2000g:03112)
  • [St] H. Steinhaus, Sur les distances des points de mesure positive, Fund. Math. 1 (1920), 93-104.
  • [THJ] F. Topsøe and J. Hoffmann-Jørgensen, Analytic spaces and their applications, Part 3 of [Rog2].
  • [Trau] R. Trautner, A covering principle in real analysis, Quart. J. Math. Oxford (2) 38 (1987), 127-130. MR 876270 (88d:26024)

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Additional Information

N. H. Bingham
Affiliation: Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom

A. J. Ostaszewski
Affiliation: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom

Keywords: Jones' theorem, Kominek's theorem, analytic set, Choquet capacity, Hamel basis, uniform convergence theorem, regular variation, automatic continuity
Received by editor(s): June 28, 2008
Received by editor(s) in revised form: April 3, 2009
Published electronically: November 5, 2009
Dedicated: To Roy Davies on the occasion of his 80th birthday
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society

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