Small Subsets of the Reals and Tree Forcing Notions

We discuss the question of which properties of smallness in the sense of measure and category (e.g. being a universally null, perfectly meager or strongly null set) imply the properties of smallness related to some tree forcing notions (e.g. the properties of being Laver-null or Miller-null). 1. Notation and terminology Most of the notation used in this paper is standard. The symbol ω denotes the set of natural numbers. By A we denote the set of all infinite sequences of elements of A (where A will always be equal to ω or 2 = {0, 1}). The symbol A stands for the set of all finite sequences of elements of A. For s ∈ A, [s] is the set of all infinite sequences of elements of A which extend s: [s] = {x ∈ A : s ⊆ x}. We say that a set T ⊆ A is a tree if T is closed under restrictions, i.e., ∀s ∈ T ∀n ≤ |s| s n ∈ T. Every tree is naturally ordered by the reverse inclusion (so s ≤ t if, and only if, s extends t, for s, t being elements of a tree). If T ⊆ A is a tree, then by [T ] we denote its body, i.e. the set of all infinite branches of T : [T ] = {x ∈ A : ∀n ∈ ω x n ∈ T }. For a tree T , we define Split(T ) to be the set {s ∈ T : |{n ∈ ω : sn ∈ T }| > 1}. By Split(T ) we will denote the n-th level of Split(T ): Split(T ) = {s ∈ Split(T ) : |{k < |s| : s k ∈ Split(T )}| = n}. Received by the editors June 11, 2002 and, in revised form, September 3, 2002. 2000 Mathematics Subject Classification. Primary 03E35, 28E15.