Convergence Theorems for Set Functions

Cafiero approach to the Dieudonné type theorems

“…Since (A n \ H n ) n∈N is a decreasing sequence of R and we have proved that (μ i ) i∈N are uniformly exhaustive, then by (2.2) of [5] we have that for every ε > 0 there exists m ∈ N such that for every n m μ i (Y ) < ε for every Y ∈ R A n \H n and i ∈ N, and this completes the proof. 2…”

“…Then G is a σ -ring so the restrictions of the μ i to G verifies the hypothesis of 2.11 of [5], then they are uniformly exhaustive and we have lim k μ i (X k ) = 0 uniformly for n ∈ N. It follows that the (μ n ) n∈N is uniformly τ -exhaustive. The thesis is a consequence of Theorem 2.8.…”

“…It is well known that the Nikodým convergence and the Nikodým boundedness theorems for measures fail, in general, in algebras of sets (see [4,5]) and that the same arrives for the Cafiero convergence theorem (see [6]). The classical Dieudonné conditions [8] ensure that the Nikodým theorems are, with suitable conditions, true in algebras.…”

On Drewnowski lemma for non-additive functions and its consequences

“…P. Cavaliere (B) Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Sa), Italy e-mail: pcavaliere@unisa.it metrizable topological group, then any disjoint sequence (a k ) k∈N in R contains a subsequence (a k i ) i∈N such that all µ n 's are σ -additive on the σ -ring generated by it [10,Proposition 2]. This result turns out to be a powerful method to lead the study of finitely additive and exhaustive functions back to that of σ -additive ones, approaching for instance Nikodym, Brooks-Jewett as well as Cafiero theorems (as shown in [6]); accordingly many authors have looked for some of its possible extensions and developments. In the Boolean context we just recall [5,20], where the assumption of the σ -completeness of the ring is dropped, whereas in a more general setting, referred to as non-commutative measure theory, we draw the reader attention to [3].…”

Cafiero and Brooks–Jewett theorems for Vitali spaces

“…Ventriglia (B) Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli Federico II, via Cintia, Complesso Monte S. Angelo, 80126 Napoli Italy e-mail: flaviaventriglia@libero.it In this note, we further investigate convergence theorems, following the line of research proposed by de Lucia and Pap in [7] and previously followed by de LuciaTraynor in [9] and de Lucia-Salvati in [8].…”