Wyniki wyszukiwania

1

On nonmeasurable selectors of countable group actions

100%

Zakrzewski P.

Fundamenta Mathematicae

|

2009

|

tom 202

|

nr 3

281-294

EN

Given a set X, a countable group H acting on it and a σ-finite H-invariant measure m on X, we study conditions which imply that each selector of H-orbits is nonmeasurable with respect to any H-invariant extension of m.

2

The uniqueness of Haar measure and set theory

100%

Zakrzewski P.

Colloquium Mathematicum

|

1997

|

tom 74

|

nr 1

109-121

EN

Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits of all points of X are uncountable. In particular, this is true if either G is a locally compact, σ-compact topological group acting continuously on X, or the space X is uniform and nonseparable, and G consists of uniformly equicontinuous unimorphisms of X.

3

Strong Fubini properties of ideals

63%

Recław I., Zakrzewski P.

Fundamenta Mathematicae

|

1999

|

tom 159

|

nr 2

135-152

EN

Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_x = {y: ⟨x,y⟩ ∈ D}$ are in J, then the sections $D^y = {x: ⟨x,y⟩ ∈ D}$ are in I for every y outside a set from J (``measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this phenomenon completely, sufficient conditions for SFP are always independent of ZFC. We show, in particular, that: • if there exists a Lusin set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a non-meager set, then ⟨MGR(X), J⟩ has SFP for every J generated by a hereditary $п^1_1$ (in the Effros Borel structure) family of closed subsets of Y (MGR(X) is the σ-ideal of all meager subsets of X), • if there exists a Sierpiński set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a set of positive outer Lebesgue measure, then $⟨NULL_μ, J⟩$ has SFP if either $J= NULL_ν$ or J is generated by any of the following families of closed subsets of Y ($NULL_μ$ is the σ-ideal of all subsets of X having outer measure zero with respect to a Borel σ-finite continuous measure μ on X): (i) all compact sets, (ii) all closed sets in $NULL_ν$ for a Borel σ-finite continuous measure ν on Y, (iii) all closed subsets of a $п^1_1$ set A ⊆ Y.

4

The existence of universal invariant measures on large sets

44%

Zakrzewski P.

Fundamenta Mathematicae

|

1989

|

tom 133

|

nr 2

113-124

Ograniczanie wyników

3 Fundamenta Mathematicae

1 Colloquium Mathematicum

4 Zakrzewski P.

1 Recław I.

1 2009

1 1999

1 1997

1 1989

On nonmeasurable selectors of countable group actions

The uniqueness of Haar measure and set theory

Strong Fubini properties of ideals

The existence of universal invariant measures on large sets