Lineability, algebrability, and sequences of random variables

Abstract

We show that, when omitting one condition in several well‐known convergence results from probability and measure theory (such as the Dominated Convergence Theorem, Fatou's Lemma, or the Strong Law of Large Numbers), we can construct “very large” (in terms of the cardinality of their systems of generators) spaces and algebras of counterexamples. Moreover, we show that on the probability space false(false[0,1false],scriptBfalse(false[0,1false]false),λfalse)$([0,1],\mathcal {B}([0,1]),\lambda )$ the families of sequences of random variables converging in probability but (i) not converging outside a set of measure 0 or (ii) not converging in arithmetic mean are also “very large”.

Lineability, differentiable functions and special derivatives

“…The first result goes deeper into the study of [33, theorem 5.2.1], where it is shown that the family D 0 is ℵ 0 -lineable. To do so, we begin by recalling a theorem from number theory and dynamical systems known as Kronecker's density theorem (see, for instance, [20,28,30]).…”

Highly tempering infinite matrices II: From divergence to convergence via Toeplitz–Silverman matrices

“…Also, let us present the notion of latticeable (introduced in [19] and later studied in [7,15]). Before that, recall that a Riesz space, also called a vector lattice, is a partially ordered (with, say, the order ) vector space X where the order structure is a lattice.…”