Additivity of Measure Implies Additivity of Category

Boolean isomorphism between partial orderings of convergent and divergent series and infinite subsets of 𝑁

“…One of the recent paradigms of set theory is the study of cardinal characteristics of the real line and structures connected with it (e.g., measure and category, the Cech-Stone remainder of N, the Baire space jV , convergent series); see [Ba,vD,FI,F2,J,M,Va]. The study of convergence went further along the lines of partial orderings of series that came from problems of real analysis; see, e.g., [Kh,V2].…”

On the cofinality of the smallest covering of the real line by meager sets. II

“…We will prove it here for completeness. We will use the following fact (see [2] or [1]): Theorem 2.11. add(yT) is the smallest size of the family F ç of such that there is no function S : co -► [co]<b) with \S(n)\ < n for all n, such that V/eF V°°« f(n) £ S(n). D Suppose that {Xa : a <k < add(yT)} is a family of C" sets.…”

Invariants of Measure and Category

“…We will establish the equivalence of (1) and (2). Suppose that X ∈ ADD(N ) and x S x is a Borel mapping.…”

“…The second part is due to Miller [29]. The first part of Theorem 4.11 was proved in [34] and the second in [2]. Todorcevic proved Theorem 4.25 [50].…”