Undominated Sequences of Integrable Functions
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Undominated Sequences of Integrable Functions Luis Bernal-Gonz´alez, Mar´ıa del Carmen Calder´on-Moreno, Marina Murillo-Arcila and Jos´e A. Prado-Bassas Abstract. 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. Mathematics Subject Classification. 15A03, 26A42, 28C15, 46A45, 46E10. Keywords. Integrable function, continuous function, undominated sequence, lineability, residual set.
1. Introduction Lebesgue’s Dominated Convergence Theorem (LDCT) is probably the most useful tool to interchange limits and integrals of a sequence of functions. In its most common version (see, e.g., [22, Chapter 1]), it asserts that if (X, M, μ) is a measure space and f, f1 , f2 , . . . are extended real-valued measurable functions on X, such that fn (x) −→ f (x) (n → ∞) for μ-almost every x ∈ X and there is an integrable function g : X → [−∞, +∞] with |fn (x)| ≤ g(x) for μ-almost every x ∈ X and all n ≥ 1, then f is integrable on X and fn − f 1 → 0 as n → ∞ (where h1 denotes the 1-norm X |h| dμ), so that, in particular, limn→∞ X fn dμ = X f dμ. The result can be generalized to extended complex-valued functions, to orders of integration p ≥ 1 and to other kinds of convergence, such as convergence in measure or μ-almost uniform convergence (see, e.g., [19, Chapter 21]), but we will focus on the former version. Since measurability of the fn s and almost everywhere pointwise convergence fn −→ f seem to be “natural” conditions in order that 1-norm convergence can take place, the following question arises: 0123456789().: V,-vol
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Is it feasible to expect fn − f 1 → 0 without assuming the existence of some dominating integrable function g? In turn, since |fn | ≤ g implies automatically integrability for the fn s and f , then, after replacing fn by fn − f , the problem can be reduced to get fn 1 → 0 by assuming fn (x) −→ 0 almost everywhere but not dominance. The aim of this paper is to provide an affirmative answer to the above question, in both topological and algebraic senses. The preliminary background and terminology is collected in Sect. 2. Our assertions, together with motivating related results in the literature, are presented in Sect. 3. Finally, the proof of our results will be provided in Sects. 4 to 6.
2. Notation and Preliminaries Those readers who are familiar with Borel measures, lineability, prevalency, and Baire categories can skip this section. As usual, we will denote by N, N0 , R, Q and c, respectively, the set of natural numbers, the set N ∪ {0}, the real line, the field of rational numbers, and the cardinality of the continuum. Assume that X is a Hausdorff topological space. Then, the family
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