On monotone hull operations

Abstract

MSC (2010) 03E15, 28A05, 03E17, 54H05We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ-ideals on the plane.

Monotone hulls for N cap M

“…"there are no monotone Borel hull operations on the ideals M, N and M ∩ N ". This answers Balcerzak and Filipczak [1,Question 23].…”

“…The order on P I (K, Σ) is defined by q ≥ p if and only if (both belong to P I (K, Σ) and) dom(q) ⊇ dom(p) and 1 Remember our convention that for x, y ∈ H(i) and c ∈ K(i) we write x ∈ Σ(c) iff x ∈ val(c), and x ∈ Σ(y) iff x = y.…”

“…The non-existence of monotone Borel hulls on I implies non-existence of such hulls on S I . While some partial results were presented in [7] and [1], not much is known about the converse implication. Problem 3.3 (Cf.…”

“…then d B ≤ do. If d B is smaller than the cofinality of the uniformity number non(I) of a Borel ideal I, then there is no monotone Borel hull operation on I (see Elekes and Máthé [7, Theorem 2.1], Balcerzak and Filipczak [1,Theorem 5]). Thus (⊗) if I is an ideal with Borel basis on R, d B < non(I) and non(I) is a regular cardinal, then there is no ⊂-monotone mapping ψ : I −→ Borel(R) ∩ I.…”

On Borel Hull Operations

“…(2) If the range of a Borel hull operation ψ consists of sets of some Borel class K, then we say that ψ is a K hull operation. By [7,2], under CH there exist monotone Borel hull operations on S I where I denotes either the ideal of Lebesgue negligible sets or the meager ideal. Adding many random or Cohen reals to a model of CH gives a model with no monotone Borel hull operations for I (where I is either the null or the meager ideal, respectively).…”