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 $([0,1],\mathcal {B}([0,1]),\lambda )$ 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”.