Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.

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From the classical notion of uniform integrability of a sequence of random variables, a new concept of integrability (called h-integrability) is introduced for an array of random variables, concerning an array of constants. We prove that this concept is weaker than other previous related notions of integrability, such as Cesàro uniform integrability [Chandra, Sankhyā Ser. A 51 (1989) 309-317], uniform integrability concerning the weights [Ordóñez Cabrera, Collect. Math. 45 (1994) 121-132] and Cesàro α-integrability [Chandra and Goswami, J. Theoret. Probab. 16 (2003) 655-669].Under this condition of integrability and appropriate conditions on the array of weights, mean convergence theorems and weak laws of large numbers for weighted sums of an array of random variables are obtained when the random variables are subject to some special kinds of dependence: (a) rowwise pairwise negative dependence, (b) rowwise pairwise non-positive correlation, (c) when the sequence of random variables in every row is ϕ-mixing. Finally, we consider the general weak * Corresponding author.E-mail address: cabrera@us.es (M. Ordóñez Cabrera).

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