Undominated Sequences of Integrable Functions

[EN] In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function.Luis Bernal-Gonzalez, Maria del Carmen Calderon-Moreno, and Jose A. Prado-Bassas have been supported by the Plan Andaluz de Investigacion de la Junta de Andalucia FQM-127 Grant P08-FQM-03543 and by MCINN Grant PGC2018-098474-B-C21. Marina Murillo-Arcila is supported by MEC, Grant MTM2016-75963-P, and PID2019-105011GB-I00.Bernal-González, L.; Calderón-Moreno, MDC.; Murillo Arcila, M.; Prado-Bassas, JA. (2020). Undominated Sequences of Integrable Functions. Mediterranean Journal of Mathematics. 17(6):1-17. https://doi.org/10.1007/s00009-020-01631-2117176Araújo, G., Bernal-González, L., Muñoz-Fernández, G.A., Prado-Bassas, J.A., Seoane-Sepúlveda, J.B.: Lineability in sequence and function spaces. Stud. Math. 237, 119–136 (2017)Aron, R.M., Bernal-González, L., Pellegrino, D.M., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016)Aron, R., García, D., Maestre, M.: Linearity in non-linear problems. RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat 95(1), 7–12 (2001)Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on $${\mathbb{R}}$$. Proc. Am. Math. Soc. 133(3), 795–803 (2005)Balcerzak, M., Bartoszewicz, A., Filipczak, M.: Nonseparable spaceability and strong algebrability of sets of continuous singular functions. J. Math. Anal. Appl. 407(2), 263–269 (2013)Bartoszewicz, A., Bienias, M., Gła̧b, S.: Lineability within Peano curves, martingales, and integral theory. J. Funct. Spaces 2018, Article ID 9762491 (2018)Bernal-González, L., Ordóñez Cabrera, M.: Lineability criteria, with applications. J. Funct. Anal. 266(6), 3997–4025 (2014). https://doi.org/10.1016/j.jfa.2013.11.014Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014)Bongiorno, B., Darji, U.B., Di Piazza, L.: Lineability of non-differentiable Pettis primitives. Monatsh. Math. 177, 345–362 (2015)Calderón-Moreno, M.C., Gerlach-Mena, P.J., Prado-Bassas, J.A.: Algebraic structure of continuous, unbounded and integrable functions. J. Math. Anal. Appl. 470, 348–359 (2019)Calderón-Moreno, M.C., Gerlach-Mena, P.J., Prado-Bassas, J.A.: Lineability and modes of convergence. RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 18 (2020)Conejero, J.A., Fenoy, M., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Lineability within probability theory settings. RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 111, 673–684 (2017)Diestel, J.: Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92. Springer, New York (1984)Enflo, P.H., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Some results and open questions on spaceability in function spaces. Trans. Am. Math. Soc. 366(2), 611–625 (2014)Gurariy, V.I., Quarta, L.: On lineability of sets of continuous functions. J. Math. Anal. Appl. 294(1), 62–72 (2004)Hinrichsen, D., Fernández, J.L.: Topología General. Urmo, Bilbao (1977)Hunt, B.R., Sauer, T., Yorke, J.A.: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Am. Math. Soc. (N.S.) 27(2), 217–238 (1992)Narici, L., Beckenstein, L.: Topological Vector Spaces, 2nd edn. CRC Press, Chapman and Hall, Boca Raton (2011)Nielsen, O.A.: An Introduction to Integration and Measure Theory, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)Oxtoby, J.C.: Measure and Category, 2nd edn. Springer, New York (1980)Rodríguez, J.: On lineability in vector integration. Mediterr. J. Math. 10, 425–9438 (2013)Rudin, W.: Real and Complex Analysis, vol. 3. McGraw-Hill Book Co., New York (1987)Seoane-Sepúlveda, J.B.: Chaos and lineability of pathological phenomena in analysis, Thesis (Ph.D.). Kent State University, ProQuest LLC, Ann Arbor, 139 (2006)Wilard, S.: General Topology. Addison Wesley, Reading (1970