Lineability and modes of convergence

Abstract

In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in L 1 -norm, are analyzed. These findings extend and complement a number of earlier results by several authors.

Lineability, algebrability, and sequences of random variables

“…Recall that the following result was recently proved in [19, Theorem 2.2], although (for the sake of completeness) we include here below a similar proof since it will be useful to understand the forthcoming Remarks 2.13 and 2.14, which are results of independent interest that rely on Theorem 2.12 ([19, Theorem 2.2]). Theorem $$\mathcal {S}_{8}$$ is strongly $$\mathfrak {c}$$‐algebrable.…”

“…Recall that the following result was recently proved in [19,Theorem 2.2], although (for the sake of completeness) we include here below a similar proof since it will be useful to understand the forthcoming Remarks 2. 13 it is straightforward to show that  ′ 8 , defined by…”

Banach spaces and Banach lattices of singular functions

“…To end this paper, we shall prove the following algebrability theorem about the convergence of sequences of S N . For recent results about lineability in function sequence spaces, see [3,14,15]. In order to study the existence of algebras, we endow the space of function sequences (R [0,1] ) N with coordenatewise multiplication.…”