Nonmeasurability with respet to Marczewski's...

Nonmeasurability with respet to Marczewski’sideal

Robert Ra lowskiWroc law University of Technology, Poland

Kyoto, 13 November 2014

Notation and Terminology

X - uncountable Polish space, Perf(X ) - all perfect subsets of XLet I ⊆ P(X ) be a fixed σ-ideal containing all singletons.

I non(I ) = min{|F | : F ⊆ X ∧ F /∈ I},I add(I ) = min{|A| : A ⊆ I ∧

⋃A /∈ I},

I cof(I ) = min{|B| : B ⊆ I ∧ (∀A ∈ I )(∃B ∈ B)A ⊆ B},I cov(I ) = min{|A| : A ⊆ I ∧

⋃A = X},

I for a fixed family of perfect subsets P ⊆ Perf (X ) letcovh(I ) = min{|A| : A ⊆ I ∧ (∃P ∈ P) P ⊆

⋃A}.

Marczewski ideal

E. Marczewski (Szpilrajn), Sur une classe de fonctions de W.Sierpinski et la classe correspondante d’ensembles, Fund. Math. 24(1935), 17–34.

Definition (Marczewski ideal s0)

Let X be any fixed uncountable Polish space. Then we say thatA ∈ P(X ) is in s0 iff

(∀P ∈ Perf (X ))(∃Q ∈ Perf (X )) Q ⊆ P ∧ Q ∩ A = ∅.

cov(s0) = covh(s0) - well known

Definition (s-measurable set)

Let X be any fixed uncountable Polish space. Then we say thatA ∈ P(X ) is s-measurable iff

(∀P ∈ Perf(X ))(∃Q ∈ Perf(X )) Q ⊆ P ∧ (Q ⊆ A ∨ Q ∩ A = ∅).

Laver tree

We say that the tree T ⊆ ω<ω is a Laver tree if

I there is a stem s ∈ T i.e for any t ∈ T t ⊆ s or s ⊆ t,

I for any t ∈ T if s ⊆ t then {n ∈ ω : tan ∈ T} ∈ [ω]ω.

For tree T define [T ] = {x ∈ ωω : (∀n ∈ ω) x � n ∈ T}.LT set of all Laver trees.

l0 ideal

Definition (ideal l0)

Let X be any fixed uncountable Polish space. Then we say thatA ∈ P(X ) is in l0 iff

(∀T ∈ LT)(∃S ∈ LT) [S ] ⊆ [T ] ∧ [S ] ∩ A = ∅.

cov(l0) = covh(l0) - well known

Definition (l-measurable set)

Let X be any fixed uncountable Polish space. Then we say thatA ∈ P(X ) is l-measurable iff

(∀T ∈ LT)(∃S ∈ LT) [S ] ⊆ [T ] ∧ ([S ] ⊆ A ∨ [S ] ∩ A = ∅).

Cardinal coefficients for s0 and l0

Theorem (Judah-Miller-Shelah, Repicky )

add(s0) ≤ cov(s0) ≤ cof (c) ≤ non(s0) = c ≤ cof (s0) ≤ 2c

Judah H., Miller A., Shelah S., Sacks forcing, Laver forcing and Martin’s

Axiom, Archive for Math Logic 31 (1992) 145-161.

Cleary ω1 ≤ add(l0) ≤ cov(l0) ≤ c holds.

Theorem (Goldstern-Repicky )

It is relatively consistent with ZFC that add(l0) < cov(l0).

Goldstern M., Repicky M., Shelah S., Spinas O., On Tree Ideals, Proc. of

the Amer. Math. Soc. vol. 123 no. 5, (1995). pp. 1573-1581.

Cardinal coefficients for s0 and l0

Theorem (Judah-Miller-Shelah, Repicky )

add(s0) ≤ cov(s0) ≤ cof (c) ≤ non(s0) = c ≤ cof (s0) ≤ 2c

Judah H., Miller A., Shelah S., Sacks forcing, Laver forcing and Martin’s

Axiom, Archive for Math Logic 31 (1992) 145-161.

Cleary ω1 ≤ add(l0) ≤ cov(l0) ≤ c holds.

Theorem (Goldstern-Repicky )

It is relatively consistent with ZFC that add(l0) < cov(l0).

Goldstern M., Repicky M., Shelah S., Spinas O., On Tree Ideals, Proc. of

the Amer. Math. Soc. vol. 123 no. 5, (1995). pp. 1573-1581.

Complete Laver tree

Tree T ⊆ ω<ω is a complete Laver tree, iff, every node t ∈ T isinfinitely splitting.The set of all complete Laver trees is denoted by the cLT.Moreover, recalling the definition of the ideal cl0, we have

Definition (ideal cl0)

We say that A ∈ P(ωω) is in cl0 iff

(∀T ∈ cLT)(∃Q ∈ cLT) [Q] ⊆ [T ] ∧ [Q] ∩ A = ∅.

Using the natural isomorphism of the perfect set P = [T ], whereT is Laver tree or complete Laver tree, with ωω, we havecovh(l0) = cov(l0) and covh(cl0) = cov(cl0).

Definition (cl-measurable set)

We say that A ∈ P(ωω) is cl-measurable iff for every completeLaver tree T ∈ cLT there is a complete Laver tree S ∈ cLT suchthat

(S ⊆ [T ] ∧ [S ] ⊆ A) ∨ ([S ] ⊆ [T ] ∧ [S ] ∩ A = ∅).

DefinitionLet C ∈ {s, l , cl} then we say that a family A of sets isC -summable iff for every subfamily B ⊆ A the union

⋃B is an

I -measurable set.

DefinitionLet X be any uncountable Polish space and let us consider acardinal κ. We say that the family A ⊆ P(X ) is κ-point family iff|{A ∈ A : x ∈ A}| < κ for all x ∈ X .

Observations

Proposition

Let X be any uncountable Polish space. Then there exists ans-summable family A ⊆ s0 s.t.

⋃A = X .

Proof.Let us enumerate X = {xξ : ξ < c} and let us defineAα = {xξ : ξ < α} ∈ s0 for any α < c. The familyA = {Aα : α < c} fulfills the assertion of the Theorem.

FactEvery Luzin and Sierpinski set is in s0.

Proposition

Let A be a c-point family consisting of Luzin subsets of X suchthat

⋃A /∈ s0. Then A is not s-summable.

Proof.Let add(A, s0) = min{|B| : B ⊆ A ∧

⋃B /∈ s0}, then

add(A, s0) = c.Assume that

⋃A is s-measurable. Find a nowhere dense set

P0 ⊆⋃A

Enumerate {Pξ : ξ < c} of all perfect subsets of P0.Let us build a sequence (Aξ, dξ) ∈ A× X for ξ < c

I Aξ ∩ Pξ 6= ∅ and dξ ∈ Pξ for every ξ < c and

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

In the α-th step of induction the familyA0 = {A ∈ A : (∃ξ < α)dξ ∈ A} has size < c. Then

⋃A0 ∈ s0

because add(A, s0) = c. Therefore we can find Aα ∈ A \ A0 suchthat Aα ∩ Pα 6= ∅. Then choose any dα ∈ Pα \

⋃{Aξ : ξ ≤ α}.

The resulting family A is not s-summable because⋃{Aξ : ξ < c}

is not s-measurable.

RemarkThis is also true if we replace a Luzin set by a Sierpinski set. Theabove Proposition also holds for l-summability.

Proposition

⋃B /∈ s0}, then

P0 ⊆⋃A

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

⋃A0 ∈ s0

⋃{Aξ : ξ ≤ α}.

Proposition

⋃B /∈ s0}, then

P0 ⊆⋃A

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

⋃A0 ∈ s0

⋃{Aξ : ξ ≤ α}.

Proposition

⋃B /∈ s0}, then

P0 ⊆⋃A

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

⋃A0 ∈ s0

⋃{Aξ : ξ ≤ α}.

Proposition

⋃B /∈ s0}, then

P0 ⊆⋃A

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

⋃A0 ∈ s0

⋃{Aξ : ξ ≤ α}.

The c-point family assumption cannot be omitted.

Proposition (CH)

There exists an s-summable family A of Luzin sets such that⋃A = R and A is not a c-point family.

Proof.Let {xξ : ξ < ω1} be an enumeration of R and let {Cξ : ξ < ω1} beany sequence of Luzin subsets of R. The familyA = {

⋃η<ξ Cη ∪ {xη} : ξ < ω1} is s-summable family of Luzin

sets and⋃A = R. Obviously A is not ω1-point family.

Abraham, U., A minimal model for ¬CH: iteration of Jensen’s reals,Transactions of the American Mathematical Society, 281(1984), 657-674.Judah H., Miller A., Shelah S., Sacks forcing, Laver forcing and Martin’sAxiom, Archive for Math Logic 31 (1992) 145-161.

Repicky M., Perfect sets and collapsing continuum, Comment. Math.

Univ. Carolin. 44,2 (2003) 315–327.

add(s0) = c = ω2

Moreover, in the Sacks model

ω1 = add(s0) < cov(s0) = c = ω2

(see proof of the Theorem 1.2 )Judah H., Miller A., Shelah S., Sacks forcing, Laver forcing andMartin’s Axiom, Archive for Math Logic 31 (1992) 145-161.

TheoremAssume that covh(s0) = c holds and let A ⊆ s0 be any c-pointfamily of subsets of the Polish space X with

⋃A /∈ s0. Then there

exists a subfamily B ⊆ A s.t.⋃

B is not s-measurable set.

Proof. Let A ⊆ s0 be an s-summable c-point family such that⋃A /∈ s0. Then, by assumption thre is a perfect nowhere dense set

P ⊆⋃A. So let {Pξ : ξ < c} be an enumeration of perfect

subsets of P. By transfinite induction we can choose(Aξ, dξ) ∈ A× X for ξ < c such that

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

TheoremAssume that covh(s0) = c holds and let A ⊆ s0 be any c-pointfamily of subsets of the Polish space X with

⋃A /∈ s0. Then there

exists a subfamily B ⊆ A s.t.⋃

B is not s-measurable set.

Proof. Let A ⊆ s0 be an s-summable c-point family such that⋃A /∈ s0. Then, by assumption thre is a perfect nowhere dense set

P ⊆⋃A. So let {Pξ : ξ < c} be an enumeration of perfect

subsets of P. By transfinite induction we can choose(Aξ, dξ) ∈ A× X for ξ < c such that

I {dξ : ξ < c} ∩⋃{Aξ : ξ < c} = ∅.

α-th step

At the α’th step of the induction, the union⋃{A ∈ A : (∃ξ < α)dξ ∈ A}

doesn’t cover Pα (as covh(s0) = c.) As such, one can find anA ∈ A such that A∩Pα 6= ∅ and A∩ {dξ : ξ < α} = ∅. After this,find d ∈ Pα \ (A ∪

⋃{Aξ : ξ < α}) and put Aα = A and dα = d .

For B = {Aξ : ξ < c} we have⋃

B is not s-measurablecontradicting the assumption that A is s-summable. �

TheoremIt is relatively consistent that there exists a partition A ∈ [s0]ω1 ofthe real line for which there exists a subfamily B ⊆ A such that⋃

B is not s-measurable and ω1 < c.

Let Pα denote the finite support iteration of Cohen forcing oflength ℵα, let P = Pω1 and let Aα = R ∩ V Pα for α < ω1 in V P.By c .c .c . argument we have in V P, R =

⋃α<ω1

Aα and|Aα| = ℵα < c = ℵω1 for all α < ω1.Consequently Aα ∈ s0 and A = {Aα+1 \ Aα : α < ω1} is apartition of R consisting of s0 sets.Then B =

⋃B is not s-summable, where

B = {Aα+1 \ Aα : α < ω1 and α odd} ⊂ A

Let P ⊆ R be an arbitrary perfect set in V P. Its Borel code belongsto V Pβ for some β < ω1. Then for every α with β ≤ α < ω1 thereis a perfect set Pα ⊆ P coded in V Pα+1 disjoint from Aα since|Aα| < ℵα+1 = cV

Pα+1 . Therefore B ∩ P 6= ∅ and B \ P 6= ∅. �

⋃α<ω1

Big point families

If X is any Polish space then a family A ⊂ P(X ) is called abig-point family iff

{x ∈ X : |{A ∈ A : x ∈ A}| < c} ∈ s0.

TheoremIf cov(s0) = c and c is a regular cardinal and A ⊆ s0 is a big-pointfamily of Marczewski null subsets of the uncountable Polish spaceX such that

{(x , y) ∈ X × X : |{A ∈ A : x , y ∈ A}| = c} ∈ s0 × s0,

then there exists a subfamily A′ ⊆ A such that⋃A′ is a Bernstein

Let {Pξ : ξ < c} be an enumeration of all perfect subsets of X . Byinduction we will find a sequence of triples(Aξ, dξ, xξ) ∈ A× X × X (ξ < c) so that for every ξ < c,

1. |{A ∈ A : dx ∈ A}| = c,

2. the set Bξ = {x ∈ X : |{A ∈ A : x , dξ ∈ A}| = c} is in s0,

3. dξ ∈ Pξ \ (⋃{Bη : η < ξ} ∪

⋃{Aη : η < ξ}),

4. xξ ∈ Pξ ∩ Aξ and {dη : η < ξ} ∩ Aξ = ∅.Assume that for some ξ ∈ c, we have constructed (Aη, dη) for eachη < ξ. By the assumptions of the theorem there is a perfect setP ′ξ ⊆ Pξ such that for all y ∈ P ′

|{A ∈ A : y ∈ A}| = c and {x ∈ X : |{A ∈ A : x , y ∈ A}| = c} ∈ s0.

Now choose dξ ∈ P ′ξ \ (

⋃{Bη : η < ξ} ∪

⋃{Aη : η < ξ}); this is

possible because cov(s0) = c. Then (1)− (3) hold. Similarly wecan choose any xξ ∈ P ′

ξ \⋃{Bη : η ≤ ξ}. Then

|{A ∈ A : xξ, dξ ∈ A}| < c for all η < ξ. By regularity of c we canchoose Aξ ∈ A so that xξ ∈ Aξ and dη /∈ Aξ for all η ≤ ξ. Thisensures (4) and the construction is finished. Now setA′ = {Aξ : ξ < c}. Then

⋃A′ is a Bernstein set because by (3)

and (4) for every ξ < c, xξ ∈ Pξ ∩⋃A′ and dξ ∈ Pξ \

⋃A′. �

MAD s0 or l0-families and their s or l-nonmeasurability

TheoremThere exists a s-nonmeasurable m.a.d. family in Baire space.

RemarkThe same Theorem is valid in case l and m nonmeasurability.

Fix T ⊆ ω < ω a perfect tree such that [T ] is a.d. in ωω. Let usenumerate Perf (T ) = {Tα : α < c} a family of all perfect subsetsof T . By transfinite reccursion let us define

{(aα, dα, xα) ∈ [T ]2 × ωω : α < c}

such that for any α < c we have:

1. aα, dα ∈ Tα,

2. {aξ : ξ < α} ∩ {dξ : ξ < α} = ∅,3. {aξ : ξ < α} ∪ {xξ : ξ < α} is a.d.,

4. ∀∞n xα(n) = dα(n) but xα 6= dα.

α-th step construction

Now assume that we are in α-th step construction and we haverequired sequence

{(aξ, dξ, xξ) ∈ [T ]2 × ωω : ξ < α}

which have size at most ω|α| < c then we can choose in [Tα] (ofsize c) aα, dα ∈ [Tα] which fulfills the first condition. Then chooseany xα ∈ ωω different than dα but (∀∞n)dα(n) = xα(n) thenx ∈ ωω \ [T ] and

{aξ : ξ < α} ∪ {xξ : ξ < α}

forms an a.d. family in ωω.

end of proof

Keppeing in mind

Now set A0 = {aα : α < c} ∪ {xα : α < c} and let us extend it toany maximal a.d. family A. It is easy to chect that A is requireds-nonmeasurable m.a.d. family in the Baire space ωω. �

cl-nonmeasurable dominating mad family

TheoremIt is relatively consistent with ZFC that

I ω1 < c,

I ∃A ⊆ ωω A is m.a.d. and is not cl-measurable,

I ∃A′ ∈ [A]<c A′ is dominating family in Baire space.

Forcing notion

Let V |= GCH.T ⊆ ω<ω Laver tree such that [T ] - a.d. family.Forcing notion (QT ,≤): p = (xp, s

gp , sbp ,Fp) ∈ QT iff

I xp ∈ ω<ω and

I sgp , sbp are nonempty finite subtrees of T (finite subsets of Tclosed under restriction),

I Fp ∈ [ωω]<ω,

For t ∈ ω<ω and nonempty finite tree τ ⊆ ω<ω lett � τ =

⋃{s ∈ τ : s ⊆ t} be the maximal initial subsequence of t

that belongs to τ .

The order is defined as follows: for every p = (xp, sgp , sbp ,Fp) ∈ QT

and q = (xq, sgq , sbq ,Fq) ∈ QT we have p ≤ q iff

1. xq ⊂ xp ∧ sgq ⊆ sgp ∧ sbq ⊆ sbp ∧ Fq ⊆ Fp,

2. (∀t ∈ sgp ) xp ∩ t ⊆ (t � sgq ) ∪ xq,

3. (∀h ∈ Fq)(∀t ∈ dom(xp) \ dom(xq)) h(t) ≤ xp(t).

4. (∀h ∈ Fq)(∀t ∈ sbp ) t ∩ h = (t � sbq ) ∩ h,

5. (∀h ∈ Fq)(∀t ∈ sgp ) t ∩ h = (t � sgq ) ∩ h.

ClaimQT is σ-centered.

Proof.Let I = {(x , sg , sb) : x ∈ ω<ω ∧ sg , sb ∈ [T ]<ω}.For every v = (x , sg , sb) ∈ I the setQv = {p ∈ QT : (xp, s

gp , sbp ) = (x , sg , sb)} is a centered subset of

QT , because for any p, q ∈ Qv the conditionr = (x , sg , sb,Fp ∪Fq) from Qv is a common extension of p and q.Since I is countable QT is σ-centered and hence it satisfies thec.c.c . .

Generic elements

Let G ⊆ QT be a generic filter over V and in V [G ] let

xG =⋃{xp : p ∈ G},

⋃{sgp : p ∈ G} and Sb

G =⋃{sbp : p ∈ G}.

cLT(T ) - collection of all complete Laver subtrees of T .

Claim(∀S ∈ cLT (T ) ∩ V ) Sg

G ∩ S ∈ cLT (T ) and SbG ∩ S ∈ cLT (T ),

Proof.Let S ∈ cLT (T ) ∩ V , t ∈ Sg

G ∩ S and n ∈ ω.We find m > n such that t_m ∈ Sg

G ∩ S .Let p ∈ G be such that t ∈ sgp . Let q ≤ p be any forcing condition.Then because S ⊆ T is a complete Laver tree one can find m ≥ nsuch that t_m ∈ S and the condition r = (xq, s

gq ∪{t_m}, sbq ,Fq)

is stronger than q. Then r t_m ∈ SgG ∩ S .

The conclusion for SbG can be proved in the same way.

ClaimFor every S ∈ cLT (T ) ∩ V the set

{z ∈ [SbG ∩ S ] : |xG ∩ z | = |xG \ z | = ω}

is comeager in [SbG ∩ S ].

For every n ∈ ω set

Un = {z ∈ [SbG ∩ S ] : |xG ∩ z | ≥ n ∧ |xG \ z | ≥ n}

is open dense in [SbG ] ∩ S ].Un open in [Sb

G ] ∩ S easy.Density:Let t ∈ Sb

G ∩ S be arbitrary, we show that there is t ′ ⊇ t s.t.[t ′] ∩ [Sb

G ] ∩ S ⊆ Un.

For every n ∈ ω set

Un = {z ∈ [SbG ∩ S ] : |xG ∩ z | ≥ n ∧ |xG \ z | ≥ n}

is open dense in [SbG ] ∩ S ].Un open in [Sb

G ] ∩ S easy.Density:Let t ∈ Sb

G ∩ S be arbitrary, we show that there is t ′ ⊇ t s.t.[t ′] ∩ [Sb

G ] ∩ S ⊆ Un.

Let p ∈ G be such that t ∈ sbp and let q ≤ p, k = max{|xq|, |t|}.We find t ′ ∈ S ∩ ωk+2n and x ∈ ωk+2n such that xq ⊆ x , t ⊆ t ′,t ′ � sgq = t, and for i < k + 2n,

1. if |t| ≤ i < k + 2n, then t ′(i) ∈ ω \ {h(i) : h ∈ Fq} andt ′ � (i + 1) ∈ S \ sbq ,

2. if |xq| ≤ i < k , then x(i) = min(ω \ {h(i) : h ∈ Fq}),

3. if k ≤ i < k + n, then x(i) = t ′(i),

4. if k + n ≤ i < k + 2n, thenx(i) = min(ω \ {t ′(i), h(i) : h ∈ Fq}).

Set r = (x , sgq , sbq ∪ {t ′ � i : i ≤ |t ′|},Fq) we have

r ≤ q ∧ r t ′ ∈ SbG ∩ S

Moreover, for any i < n

r (∀z ∈ [t ′])xG (k + i) = z(k + i) and xG (k +n + i) 6= z(k +n + i)

thenr [t ′] ∩ [Sb

G ∩ S ] ⊆ Un

3. if k ≤ i < k + n, then x(i) = t ′(i),

r ≤ q ∧ r t ′ ∈ SbG ∩ S

G ∩ S ] ⊆ Un

3. if k ≤ i < k + n, then x(i) = t ′(i),

r ≤ q ∧ r t ′ ∈ SbG ∩ S

G ∩ S ] ⊆ Un

3. if k ≤ i < k + n, then x(i) = t ′(i),

r ≤ q ∧ r t ′ ∈ SbG ∩ S

G ∩ S ] ⊆ Un

ClaimIf F ⊆ ωω ∩ V is an almost disjoint family, then also the families{xG} ∪ [Sg

G ] ∪ F and [SbG ] ∪ F are also almost disjoint.

ClaimxG is dominating in ωω ∩ V .

For ω1 ≤ cof (κ) < κ, let Pκ fs-iteration of QT forcings of length κ.For β < κ xH ∈ V Pβ+1 name for H generic real where H ⊆ QT

over V Pβ

xHβ= (xH)Gβ+1

and similarly SgHβ

, SbHβ

Let A = ∅ and

I Aβ+1 = Aβ ∪ {xHβ} ∪ [Sg

I γ-limit then Aγ =⋃ξ<β Aξ

Let B such that A ⊆ B and B - ⊆-maximal a.d. family. B fulfillsassertion.

over V Pβ

xHβ= (xH)Gβ+1

and similarly SgHβ

, SbHβ

Let A = ∅ and

over V Pβ

xHβ= (xH)Gβ+1

and similarly SgHβ

, SbHβ

Let A = ∅ and

Thank You

Marczewski (Szpilrajn) E., Sur une classe de fonctions de W.Sierpinski et la classe correspondante d’ensembles, Fund.Math. 24 (1935), 17–34.

Abraham, U., A minimal model for ¬CH: iteration of Jensen’sreals, Transactions of the American Mathematical Society,281(1984), 657-674.

Goldstern M., Repicky M., Shelah S., Spinas O., On TreeIdeals, Proc. of the Amer. Math. Soc. vol. 123 no. 5, (1995).pp. 1573-1581.

Jech T., Set theory, millenium edition, Springer Monographs inMathematics, Springer-Verlag, (2003).

Judah H., Miller A., Shelah S., Sacks forcing, Laver forcing andMartin’s Axiom, Archive for Math Logic 31 (1992) 145-161.

Kunen, K., Set Theory. An Introduction to IndependenceProofs, North Holland, Amsterdamm, New York, Oxford 1980.

Repicky M., Perfect sets and collapsing continuum, Comment.Math. Univ. Carolin. 44,2 (2003) 315–327.

Yorioka T., Forcings with the countable chain condition andthe covering number of the Marczewski ideal, Arch. Math.Logic 42 (2003), 695–710.

R. Ra lowski and Sz. Zeberski, Group action on Polish spaces,arXiv:1406.3063

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