Some Queries Concerning Convergence Theorems

Abstract

In this query we ask if the following 11 classical theorems on convergence are equivalent: the Lebesgue-Beppo Levi Theorem \cite[p. 141]{N1}, a theorem on the integration of a series with positive terms \cite[p. 142]{N1}, the Fatou Lemma I \cite[p. 172]{H12}, the Fatou Lemma II \cite[p. 140]{N1}, the Fatou Lemma III \cite[p. 140]{N1}, Lebesgue's Dominated Convergence Theorem I \cite[p. 172]{H12}, Lebesgue's Dominated Convergence Theorem II \cite[p. 173]{H12}, Lebesgue's Dominated Convergence Theorem III \cite[pp. 149-50]{N1}, Vitali's Theorem \cite[p. 152]{N1}, Lebesgue's Dominated Convergence Theorem for Bounded Functions I \cite[p. 127]{N1}, Lebesgue's Dominated Convergence Theorem for Bounded Functions II.