The class of perfectly null sets Preliminaries and introduction - PowerPoint PPT Presentation

The class of perfectly null sets and its transitive version Micha� l Korch The class of perfectly null sets Preliminaries and introduction and its transitive version Perfectly null sets Perfectly null sets in the transitive sense Micha� l Korch joint work with T. Weiss Faculty of Mathematics, Informatics, and Mechanics University of Warsaw Hejnice, February 2015

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and introduction Perfectly null sets universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure sense

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and introduction Perfectly null sets universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure sense strongly null SN can be covered by a sequence of open sets of any given sequence of diameters

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and introduction Perfectly null sets universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure sense strongly null SN Thm. (Galvin-Mycielski-Solovay). Iff it can be shifted away from any meager set.

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and introduction Perfectly null sets universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure sense strongly null SN strongly meager SM can be covered by sequence of open sets of any given can be shifted away from any null set sequence of diameters

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and � introduction Perfectly null sets universally meager UM universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure its every Borel isomorphic image is meager in 2 ω [1], [9] sense strongly null SN strongly meager SM can be covered by sequence of open sets of any given can be shifted away from any null set sequence of diameters

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive version meager in any perfect set (in the subspace topology) [2] Micha� l Korch Preliminaries and � introduction Perfectly null sets universally meager UM universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure its every Borel isomorphic image is meager in 2 ω [1], [9] sense � perfectly meager in the transitive sense PM’ will be defined later [4] � strongly null SN strongly meager SM can be covered by sequence of open sets of any given can be shifted away from any null set sequence of diameters

Preliminaries: special subsets in 2 ω The class of perfectly null sets perfectly meager PM and its transitive perfectly null PN version meager in any perfect set (in the subspace topology) [2] our aim Micha� l Korch Preliminaries and � � ? introduction Perfectly null sets universally meager UM universally null UN Perfectly null sets in the transitive null with respect to any finite Borel diffused measure its every Borel isomorphic image is meager in 2 ω [1], [9] sense � ? � perfectly null perfectly meager in the transitive sense PN’ in the transitive sense PM’ our aim will be defined later [4] � ? � strongly null SN strongly meager SM can be covered by sequence of open sets of any given can be shifted away from any null set sequence of diameters

Outline The class of perfectly null sets and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets 1 Perfectly null sets definitions Perfectly null sets simple properties in the transitive sense main open problem Perfectly null in the transitive sense sets 2 definitions two theorems open problems

Perfectly null sets: measure on perfect sets The class of perfectly null sets and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense

Perfectly null sets: measure on perfect sets The class of perfectly null sets and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense

Perfectly null sets: measure on perfect sets The class of perfectly null sets Canonical homeomorphism: h P : 2 ω → P and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense

Perfectly null sets: measure on perfect sets The class of perfectly null sets Canonical homeomorphism: h P : 2 ω → P and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense Measure on pefect set µ P ( A ) = λ ( h − 1 P [ A ]) , where λ is the standard Lebesgue measure on 2 ω .

Perfectly null sets: measure on perfect sets The class of Canonical homeomorphism: h P : 2 ω → P perfectly null sets and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense Measure on pefect set µ P ( A ) = λ ( h − 1 P [ A ]) , where λ is the standard Lebesgue measure on 2 ω . Ex. µ P ([110]) = 1 4 .

Perfectly null sets: measure on perfect sets The class of Canonical homeomorphism: h P : 2 ω → P perfectly null sets and its transitive version Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense Measure on pefect set µ P ( A ) = λ ( h − 1 P [ A ]) , where λ is the standard Lebesgue measure on 2 ω . Ex. µ P ([110]) = 1 4 .

Perfectly null sets The class of perfectly null sets and its transitive version Micha� l Korch Definition A set X ⊆ 2 ω is perfectly null if for every perfect set P ⊆ 2 ω , Preliminaries and introduction µ P ( P ∩ X ) = 0. Perfectly null sets Perfectly null sets in the transitive sense

Perfectly null sets The class of perfectly null sets and its transitive version Micha� l Korch Definition A set X ⊆ 2 ω is perfectly null if for every perfect set P ⊆ 2 ω , Preliminaries and introduction µ P ( P ∩ X ) = 0. Perfectly null sets Perfectly null sets in the transitive sense Observation UN ⊆ PN.

Perfectly null sets The class of perfectly null sets and its transitive version Micha� l Korch Definition A set X ⊆ 2 ω is perfectly null if for every perfect set P ⊆ 2 ω , Preliminaries and introduction µ P ( P ∩ X ) = 0. Perfectly null sets Perfectly null sets in the transitive sense Observation UN ⊆ PN. Recall that a set X is in Marczewski ideal s 0 if for any perfect set P , there exists a perfect set Q ⊆ P such that X ∩ Q = ∅ .

Perfectly null sets The class of perfectly null sets and its transitive version Micha� l Korch Definition A set X ⊆ 2 ω is perfectly null if for every perfect set P ⊆ 2 ω , Preliminaries and introduction µ P ( P ∩ X ) = 0. Perfectly null sets Perfectly null sets in the transitive sense Observation UN ⊆ PN. Recall that a set X is in Marczewski ideal s 0 if for any perfect set P , there exists a perfect set Q ⊆ P such that X ∩ Q = ∅ . Observation PN ⊆ s 0 .

The main problem The class of The main open question perfectly null sets and its transitive version Is it consistent, that UN � PN? Micha� l Korch Preliminaries and introduction Perfectly null sets Perfectly null sets in the transitive sense

The main problem The class of The main open question perfectly null sets and its transitive version Is it consistent, that UN � PN? Micha� l Korch On the category side all known arguments proving that it is consistent Preliminaries and introduction that UM � PM use the idea of the Lusin function or similar ideas. Perfectly null sets Perfectly null sets Lusin function (Lusin, Sierpi´ nski, [7]) in the transitive sense There exists a function L : ω ω → 2 ω , such that: L is continuous and one-to-one, if L is a Lusin set, then L [ L ] ∈ PM, L − 1 is of the Baire class one.