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

Definition. A tree T ⊆ ω<ω is called:

• 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.

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Definition. We say that a set X ⊆ ωω has:

• s0–property if

∀T ∈ S ∃T ′ ∈ S T ′ ⊆ T ∧ [T ′] ∩ X = ∅,

• m0–property if

∀T ∈ M ∃T ′ ∈ M T ′ ⊆ T ∧ [T ′] ∩ X = ∅,

• l0–property if

∀T ∈ L ∃T ′ ∈ L T ′ ⊆ T ∧ [T ′] ∩ X = ∅, It is known that none of these properties imply any other.

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Small sets in the sense of measure and category: Null sets

• Meager sets

Perfectly meager sets

• Universally

null sets

• Universally

meager sets

• Very

meager sets

• Strongly

null sets

• Strongly

meager sets

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Definition. A set X ⊆ ωω is called:

• universally null, if it has measure zero with

respect to any Borel probabilistic measure

• n ωω vanishing on points.
• perfectly meager, if for every perfect set

P ⊆ ωω the set X ∩ P is meager in P as a topological subspace of ωω. Theorem.

• Every universally null set has s0–property,
• Every perfectly meager set has s0–property.

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Definition. A set X ⊆ 2ω is called:

• strongly meager, if for every G ∈ N ∗ exists

t ∈ 2ω such that X ⊆ t + G.

• very meager, if for every G ∈ N ∗ exists a

countable set T ⊆ 2ω such that X ⊆ T + G. Obviously, every strongly meager set is very meager. Theorem. (Nowik–Weiss) Every very mea- ger set X ⊆ 2ω has both m0– and l0–property.

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Theorem. Every perfectly meager set X ⊆ ωω has m0–property. Lemma. For every meager set F ⊆ ωω exists a Miller tree T such that [T] ∩ F = ∅.

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Definition. A set X ⊆ ωω is universally meager if for every Borel isomorphism f : ωω → ωω the set f[X] is meager in ωω. It is easy to check that every universally mea- ger set is perfectly meager. Theorem. Under CH not every universally mea- ger set has l0–property.

• Proof. An ω1-scale is a strictly increasing (in

the sense of ∗) and dominating sequence of elements of ωω of length ω1. Step 1: Every ω1–scale is universally meager. Step 2: Under CH, there exists an ω1–scale which intersects [T] for every Laver tree T.

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Proposition. It is consistent that every per- fectly meager set has l0–property. Proof.

• (Miller) Consistently, every perfectly mea-

ger set has size smaller than continuum.

• Every set of cardinality smaller than con-

tinuum has l0–property.

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Definition. A set X ⊆ ωω is called strongly null, if for every sequence εn : n ∈ ω of positive real numbers exists a sequence of open sets In : n ∈ ω such that diam(In) < εn and X ⊆

• n∈ω

In. Question. Does every strongly null set have l0– and/or m0–property? It turns out that the answer depends on the choice of a metric on ωω.

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Let the metric d be defined as follows: d(f, g) = 1 min{n ∈ ω : f(n) = g(n)}. Theorem. If a set X ⊆ ωω is strongly null in ωω with the metric d then it has l0–property. Theorem. Under CH there exists a set X ⊆ 2ω which is strongly null but does not have l0–property. Proof. Step 1. We have already shown how to con- struct an ω1-scale which does not have l0– property. Step 2. It is well known that every ω1–scale is strongly null as a subset of 2ω.

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We have a very similar situation with m0–property. Theorem. If a set X ⊆ ωω is strongly null in ωω with the metric d then it has a m0–property. Lemma. For every countable family F ⊆ ωω exists a strictly increasing g ∈ ωω such that for every f ∈ F and for every injection i : ω → ω we have g ◦ i ∗ f.

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Theorem. Under CH not every strongly null subset of 2ω has m0–property.

• Proof. Assume CH. Let Tα : α < ω1 be an

enumeration of all Miller trees and let fα : α < ω1 be an enumeration of ωω. Construct an increasing sequence Mα : α < ω1 of countable transitive models of ZFC∗ such that fα, Tα ∈ Mα. Let Gα be a M-generic over Mα such that Tα ∈ Gα and let xα ∈ {[T] : T ∈ Gα}. The set X = {xα : α < ω1} is the one we are looking for. Step 1: X does not have m0–property, because it intersects [T] for every Miller tree T. Step 2: X is strongly null, because it is con- centrated on Q ⊆ 2ω (i.e. X \ U is countable for every open set U ⊇ Q).

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Proposition. Under CH:

• not every universally null set has m0–property,
• not every universally null set has l0–property.

Proof. It follows from the fact that every strongly null subset of 2ω is universally null.

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Proposition. It is consistent that every uni- versally null set has both m0– and l0–property. Proof.

• (Miller) Consistently, every universally null

set has cardinality smaller than continuum.

• Every set of cardinality smaller than con-

tinuum has both l0– and m0–property.

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

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