The Nikodym property and cardinal characteristics of the continuum

Convergence of measures in forcing extensions

“…Obviously, a Boolean algebra has the Nikodym property if it does not have any anti-Nikodym sequences of measures. The following lemma is an important tool in studying the Nikodym property -see Sobota [28,Lemmas 4.4 and 4.7] for a proof. Lemma 2.6.…”

“…In this section we provide several consequences of Theorem 5.3 concerning cardinal characteristics of the continuum and the Efimov problem. Since every countable Boolean algebra has neither the Nikodym property nor the Grothendieck property and ℘(ω) has both of the properties, we immediately get that ω 1 ≤ nik, gr ≤ vhs ≤ c. The relations between nik and other classical cardinal characteristics of the continuum were studied in Sobota [28], where the following inequalities were proved:…”

“…( Inequalities (1) and (4) may suggest that the dominating number d is a good candidate for bounding nik and gr from below (e.g. Sobota [28,Question 3.5]). However, using the countable support iteration of length ω 2 of Miller's forcing, we obtain the model where ω 1 < d = ω 2 = c (see Blass [4,Section 11.9]) and so, by Corollary 6.2, the following holds.…”

“…Boolean algebras with the Nikodym property or the Grothendieck property yield examples of Efimov spaces -it is well-known that if a Boolean algebra A has either the Nikodym property or the Grothendieck property, then its Stone space K A does not have any non-trivial convergent sequences, and hence if ω < |A| < c, then K A is an Efimov space (since w K A = |A|). In Sobota [28,Section 8.2], assuming that cof(N ) ≤ κ = cof [κ] ω , ⊆ < c, a Boolean algebra with the Nikodym property and of cardinality κ was constructed, so an Efimov space of weight κ was obtained as well. We can apply this argument here -together with Theorem 5.3 -to prove the following corollary.…”

“…Recently, Sobota and Zdomskyy [29] proved the same result for the Nikodym property, whichtogether with Brech's result -consistently yields a class of examples of Boolean algebras with the Vitali-Hahn-Saks property and of cardinality ω 1 while the inequality ω 1 < c holds true in the model. Also, Sobota [28] proved in ZFC that for every cardinal number κ such that cof([κ] ω ) = κ ≥ cof(N ) (the cofinality of the Lebesgue null ideal) there exists a Boolean algebra with the Nikodym property and of cardinality κ.…”