Introduction Uniformity and covering number Cofinality Questions Cardinal invariants of the Haar null ideal M´ ark Po´ or E¨ otv¨ os Lor´ and University, Budapest Descriptive Set Theory in Turin September 2017 joint work with M´ arton Elekes M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Introduction The Haar null ideal HN ( G ). Definition Let G be a Polish group, N ⊆ G be a set. Then N is Haar null, N ∈ HN ( G ) if there exists a Borel set B ⊇ N, and a probability Borel measure µ on G such that ∀ g , h ∈ G µ ( hBg ) = 0 . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Introduction The Haar null ideal HN ( G ). Definition Let G be a Polish group, N ⊆ G be a set. Then N is Haar null, N ∈ HN ( G ) if there exists a Borel set B ⊇ N, and a probability Borel measure µ on G such that ∀ g , h ∈ G µ ( hBg ) = 0 . The original notion HN UM ( G ), in the sense of Christensen: Definition Let G be a Polish group, N ⊆ G be a set. Then N is generalized Haar null, N ∈ HN UM ( G ) if there exists a universally measurable set U ⊇ N, and a probability Borel measure µ on G such that ∀ g , h ∈ G µ ( hUg ) = 0 . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts It is a generalisation of the null ideal: Theorem If G is a locally compact Polish group, then HN ( G ) = HN UM ( G ) , moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts It is a generalisation of the null ideal: Theorem If G is a locally compact Polish group, then HN ( G ) = HN UM ( G ) , moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure. Theorem If G is a Polish group then HN ( G ) and HN UM ( G ) form σ -ideals. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts It is a generalisation of the null ideal: Theorem If G is a locally compact Polish group, then HN ( G ) = HN UM ( G ) , moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure. Theorem If G is a Polish group then HN ( G ) and HN UM ( G ) form σ -ideals. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group that admits an invariant metric. Then there is a set C ∈ HN UM ( G ) ∩ Π 1 1 such that C / ∈ HN ( G ) . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts It is a generalisation of the null ideal: Theorem If G is a locally compact Polish group, then HN ( G ) = HN UM ( G ) , moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure. Theorem If G is a Polish group then HN ( G ) and HN UM ( G ) form σ -ideals. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group that admits an invariant metric. Then there is a set C ∈ HN UM ( G ) ∩ Π 1 1 such that C / ∈ HN ( G ) . i.e. ∃ µ probability Borel measure s.t. µ ∗ ( gCh ) = 0 ( ∀ g , h ) , ∄ ν, B ∈ ∆ 1 1 such that C ⊆ B, and ν ( gBh ) = 0 ( ∀ g , h ) . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions A sufficient condition for non-Haar nullness Definition In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g , h such that C ⊆ g − 1 Sh − 1 ) gCh ⊆ S ( ⇐ ⇒ M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions A sufficient condition for non-Haar nullness Definition In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g , h such that C ⊆ g − 1 Sh − 1 ) gCh ⊆ S ( ⇐ ⇒ A compact-catcher (Borel set) B cannot be Haar null. If µ is an arbitrary probability Borel measure (on a Polish group G ), then (by regularity) there is a compact set C with µ ( C ) > 0. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions A sufficient condition for non-Haar nullness Definition In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g , h such that C ⊆ g − 1 Sh − 1 ) gCh ⊆ S ( ⇐ ⇒ A compact-catcher (Borel set) B cannot be Haar null. If µ is an arbitrary probability Borel measure (on a Polish group G ), then (by regularity) there is a compact set C with µ ( C ) > 0. Then translating B so that it covers C C ⊆ gBh ⇒ 0 < µ ( C ) ≤ µ ( gBh ) . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts A Haar null set B ∈ ∆ 1 1 does not necessarily have a G δ hull, in fact the following holds. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω 1 . Then there exists a Haar null Borel set B α ∈ HN ( G ) such that there is no B ′ ∈ HN ( G ) with B α ⊆ B ′ and B ′ ∈ Π 0 α . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts A Haar null set B ∈ ∆ 1 1 does not necessarily have a G δ hull, in fact the following holds. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω 1 . Then there exists a Haar null Borel set B α ∈ HN ( G ) such that there is no B ′ ∈ HN ( G ) with B α ⊆ B ′ and B ′ ∈ Π 0 α . Corollary If G is non-locally compact with an invariant metric then add( HN ( G )) = ω 1 . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts A Haar null set B ∈ ∆ 1 1 does not necessarily have a G δ hull, in fact the following holds. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω 1 . Then there exists a Haar null Borel set B α ∈ HN ( G ) such that there is no B ′ ∈ HN ( G ) with B α ⊆ B ′ and B ′ ∈ Π 0 α . Corollary If G is non-locally compact with an invariant metric then add( HN ( G )) = ω 1 . Proof. Let B α -s ( α < ω 1 ) be given by the theorem. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Facts A Haar null set B ∈ ∆ 1 1 does not necessarily have a G δ hull, in fact the following holds. Theorem (M. Elekes, Z. Vidny´ anszky) Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω 1 . Then there exists a Haar null Borel set B α ∈ HN ( G ) such that there is no B ′ ∈ HN ( G ) with B α ⊆ B ′ and B ′ ∈ Π 0 α . Corollary If G is non-locally compact with an invariant metric then add( HN ( G )) = ω 1 . Proof. Let B α -s ( α < ω 1 ) be given by the theorem. Now if B ⊇ � α<ω 1 B α is a Borel Haar null set, and B ∈ Π 0 β for some countable β , B β ⊆ B is a contradiction. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Uniformity and covering number Theorem (T. Banakh) cov( HN UM ( Z ω )) = min( b , cov( N )) non( HN UM ( Z ω )) = max( d , non( N )) M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Uniformity and covering number Theorem (T. Banakh) cov( HN UM ( Z ω )) = min( b , cov( N )) non( HN UM ( Z ω )) = max( d , non( N )) The theorem also holds in a bit more general setting, as does the following, which is obtained by a minor modification of Banakh’s proof. M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Uniformity and covering number Theorem (T. Banakh) cov( HN UM ( Z ω )) = min( b , cov( N )) non( HN UM ( Z ω )) = max( d , non( N )) The theorem also holds in a bit more general setting, as does the following, which is obtained by a minor modification of Banakh’s proof. Theorem (M. Elekes, M.P.) cov( HN ( Z ω )) = min( b , cov( N )) non( HN ( Z ω )) = max( d , non( N )) M´ ark Po´ or Cardinal invariants of the Haar-null ideal
Introduction Uniformity and covering number Cofinality Questions Cofinality Theorem (T. Banakh) Let G be a non-locally compact Polish group admitting an invariant metric. Then cof( HN UM ( G )) > min( d , non( N )) . M´ ark Po´ or Cardinal invariants of the Haar-null ideal
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