# Lineability, algebrability, and sequences of random variables

### Impacto

Fernández Sánchez, Juan and Seoane-Sepúlveda, Juan B. and Trutschnig, Wolfgang (2022) Lineability, algebrability, and sequences of random variables. Mathematische Nachrichten, 295 (5). pp. 861-875. ISSN 0025-584X

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Official URL: https://doi.org/10.1002/mana.202000102

## 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”.

Item Type: Article Lineability; Algebrability; Vector series; Probability theory; Random variable; Stochastic process 74695 22 Sep 2022 11:27 22 Sep 2022 11:31