Various classes of small sets: combinatorics vs. measure and - PDF document

Various classes of small sets: combinatorics vs. measure and category Marcin Kysiak (joint work with Tomasz Weiss) London, April 3rd, 2002

A tree T ⊆ ω <ω is called: Definition. • a perfect tree if ∀ s ∈ T ∃ t ∈ T t ⊇ s ∧ |{ n ∈ ω : t ⌢ n ∈ T }| > 1 , • a Miller tree (or a superperfect tree) if ∀ s ∈ T ∃ t ∈ T t ⊇ s ∧ |{ n ∈ ω : t ⌢ n ∈ T }| = ω, • a Laver tree if ∃ s ∈ T ∀ t ∈ T t ⊆ s ∨ |{ n ∈ ω : t ⌢ n ∈ T }| = ω. By S , M , L we will denote collections of perfect, Miller and Laver trees, respectively. 1

We say that a set X ⊆ ω ω has: Definition. • s 0 –property if ∀ T ∈ S ∃ T ′ ∈ S T ′ ⊆ T ∧ [ T ′ ] ∩ X = ∅ , • m 0 –property if ∀ T ∈ M ∃ T ′ ∈ M T ′ ⊆ T ∧ [ T ′ ] ∩ X = ∅ , • l 0 –property if ∀ T ∈ L ∃ T ′ ∈ L T ′ ⊆ T ∧ [ T ′ ] ∩ X = ∅ , It is known that none of these properties imply any other. 2

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Small sets in the sense of measure and category: Meager sets Null sets � � � � � � � � � � � � � Perfectly meager sets � � � � � � � � � � � � Universally null sets � � � � � � � � � � � � Universally meager sets Very meager sets � � � � � � � � � � � � � � Strongly null sets � � � � � � � � � � � � � � Strongly meager sets 3

A set X ⊆ ω ω is called: Definition. • universally null, if it has measure zero with respect to any Borel probabilistic measure on ω ω vanishing on points. • perfectly meager, if for every perfect set P ⊆ ω ω the set X ∩ P is meager in P as a topological subspace of ω ω . Theorem. • Every universally null set has s 0 –property, • Every perfectly meager set has s 0 –property. 4

A set X ⊆ 2 ω is called: Definition. • strongly meager, if for every G ∈ N ∗ exists t ∈ 2 ω such that X ⊆ t + G . • very meager, if for every G ∈ N ∗ exists a countable set T ⊆ 2 ω such that X ⊆ T + G . Obviously, every strongly meager set is very meager. Theorem. (Nowik–Weiss) Every very mea- ger set X ⊆ 2 ω has both m 0 – and l 0 –property. 5

Theorem. Every perfectly meager set X ⊆ ω ω has m 0 –property. For every meager set F ⊆ ω ω exists Lemma. a Miller tree T such that [ T ] ∩ F = ∅ . 6

A set X ⊆ ω ω is universally meager Definition. if for every Borel isomorphism f : ω ω → ω ω the set f [ X ] is meager in ω ω . It is easy to check that every universally mea- ger set is perfectly meager. Theorem. Under CH not every universally mea- ger set has l 0 –property. Proof. An ω 1 -scale is a strictly increasing (in the sense of � ∗ ) and dominating sequence of elements of ω ω of length ω 1 . Step 1: Every ω 1 –scale is universally meager. Step 2: Under CH, there exists an ω 1 –scale which intersects [ T ] for every Laver tree T . � 7

Proposition. It is consistent that every per- fectly meager set has l 0 –property. Proof. • (Miller) Consistently, every perfectly mea- ger set has size smaller than continuum. • Every set of cardinality smaller than con- tinuum has l 0 –property. � 8

A set X ⊆ ω ω is called strongly null, Definition. if for every sequence � ε n : n ∈ ω � of positive real numbers exists a sequence of open sets � I n : n ∈ ω � such that diam ( I n ) < ε n and � X ⊆ I n . n ∈ ω Question. Does every strongly null set have l 0 – and/or m 0 –property? It turns out that the answer depends on the choice of a metric on ω ω . 9

Let the metric d be defined as follows: 1 d ( f, g ) = min { n ∈ ω : f ( n ) � = g ( n ) } . If a set X ⊆ ω ω is strongly null in Theorem. ω ω with the metric d then it has l 0 –property. Theorem. Under CH there exists a set X ⊆ 2 ω which is strongly null but does not have l 0 –property. Proof. Step 1. We have already shown how to con- struct an ω 1 -scale which does not have l 0 – property. Step 2. It is well known that every ω 1 –scale is strongly null as a subset of 2 ω . � 10

We have a very similar situation with m 0 –property. If a set X ⊆ ω ω is strongly null in Theorem. ω ω with the metric d then it has a m 0 –property. For every countable family F ⊆ ω ω Lemma. exists a strictly increasing g ∈ ω ω such that for every f ∈ F and for every injection i : ω → ω we have g ◦ i � � ∗ f. 11

Theorem. Under CH not every strongly null subset of 2 ω has m 0 –property. Proof. Assume CH. Let � T α : α < ω 1 � be an enumeration of all Miller trees and let � f α : α < ω 1 � be an enumeration of ω ω . Construct an increasing sequence � M α : α < ω 1 � of countable transitive models of ZFC ∗ such that f α , T α ∈ M α . Let G α be a M -generic over M α such that T α ∈ G α and let x α ∈ � { [ T ] : T ∈ G α } . The set X = { x α : α < ω 1 } is the one we are looking for. Step 1: X does not have m 0 –property, because it intersects [ T ] for every Miller tree T . Step 2: X is strongly null, because it is con- centrated on Q ⊆ 2 ω (i.e. X \ U is countable for every open set U ⊇ Q ). � 12

Proposition. Under CH: • not every universally null set has m 0 –property, • not every universally null set has l 0 –property. Proof. It follows from the fact that every strongly null subset of 2 ω is universally null. � 13

Proposition. It is consistent that every uni- versally null set has both m 0 – and l 0 –property. Proof. • (Miller) Consistently, every universally null set has cardinality smaller than continuum. • Every set of cardinality smaller than con- tinuum has both l 0 – and m 0 –property. � 14

These slides are already available on my web- page: http://www.impan.gov.pl/~mkysiak/ The preprint should appear there very soon as well. 15