Preliminaries Problems An example Sequential properties of measures Piotr Borodulin–Nadzieja (Wroc� law) Winterschool 2011, Hejnice joint work with Omar Selim (Norwich) Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Space of probability measures Notation K - a (Hausdorff) compact space; N = { 1 , 2 , . . . } ; P ( K ) - space of probability Borel measures on K . Weak* convergence A sequence ( µ n ) from P ( K ) is weak ∗ convergent to µ if � � f d µ n → f d µ K K for each continuous f : K → R . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Space of probability measures Notation K - a (Hausdorff) compact space; N = { 1 , 2 , . . . } ; P ( K ) - space of probability Borel measures on K . Weak* convergence A sequence ( µ n ) from P ( K ) is weak ∗ convergent to µ if � � f d µ n → f d µ K K for each continuous f : K → R . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Weak* convergence in 0-dim spaces Weak* convergence A sequence ( µ n ) from P ( K ) is weak ∗ convergent to µ if � � f d µ n → f d µ K K for each continuous f : K → R . Remark If K is zero–dimensional, then µ n converges weakly to µ if and only if µ n ( A ) → µ ( A ) for every clopen subset A ⊆ K . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Levels of complexity in P(K) Sequential closures h : K → h [ K ] ⊆ P ( K ) defined by h ( x ) = δ x is a homeomorphism; S 0 ( K ) = conv ( { δ x : x ∈ K } ); let S 1 ( K ) be the weak ∗ –sequential closure of S 0 ( K ); generally: let S α ( K ) be the weak ∗ –sequential closure of � β<α S β ( K ); S ( K ) = S ω 1 ( K ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Levels of complexity in P(K) Sequential closures h : K → h [ K ] ⊆ P ( K ) defined by h ( x ) = δ x is a homeomorphism; S 0 ( K ) = conv ( { δ x : x ∈ K } ); let S 1 ( K ) be the weak ∗ –sequential closure of S 0 ( K ); generally: let S α ( K ) be the weak ∗ –sequential closure of � β<α S β ( K ); S ( K ) = S ω 1 ( K ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Levels of complexity in P(K) Sequential closures h : K → h [ K ] ⊆ P ( K ) defined by h ( x ) = δ x is a homeomorphism; S 0 ( K ) = conv ( { δ x : x ∈ K } ); let S 1 ( K ) be the weak ∗ –sequential closure of S 0 ( K ); generally: let S α ( K ) be the weak ∗ –sequential closure of � β<α S β ( K ); S ( K ) = S ω 1 ( K ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Levels of complexity in P(K) Sequential closures h : K → h [ K ] ⊆ P ( K ) defined by h ( x ) = δ x is a homeomorphism; S 0 ( K ) = conv ( { δ x : x ∈ K } ); let S 1 ( K ) be the weak ∗ –sequential closure of S 0 ( K ); generally: let S α ( K ) be the weak ∗ –sequential closure of � β<α S β ( K ); S ( K ) = S ω 1 ( K ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example A measure outside the sequential closure Remark If µ ∈ S ( K ), then it has a separable carrier, i.e. a closed set F ⊆ K with µ ( F ) = 1 (not necessarily the support). Corollary Let R = Bor ([0 , 1]) / Null be the measure algebra and let R be its Stone space. Then the standard measure λ on R is in P ( R ) but not in S ( R ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example A measure outside the sequential closure Remark If µ ∈ S ( K ), then it has a separable carrier, i.e. a closed set F ⊆ K with µ ( F ) = 1 (not necessarily the support). Corollary Let R = Bor ([0 , 1]) / Null be the measure algebra and let R be its Stone space. Then the standard measure λ on R is in P ( R ) but not in S ( R ). Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Uniform distribution Fact A measure µ is in S 1 ( K ) if and only if it has a uniformly distributed sequence. Theorems Many spaces K have property: P ( K ) = S 1 ( K ). E.g. scattered spaces; metric spaces; 2 ω 1 [Losert, 79]; 2 c [Fremlin, 00’s]. Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Uniform distribution Fact A measure µ is in S 1 ( K ) if and only if it has a uniformly distributed sequence. Theorems Many spaces K have property: P ( K ) = S 1 ( K ). E.g. scattered spaces; metric spaces; 2 ω 1 [Losert, 79]; 2 c [Fremlin, 00’s]. Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Problems Theorem (Plebanek, PBN) If K is Koppelberg compact, then P ( K ) = S ( K ). Problem 1 Is there a space K such that S 1 ( K ) � = S ( K )? Problem 2 Is there a space K such that S 1 ( K ) � = S ( K ) = P ( K )? Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Problems Theorem (Plebanek, PBN) If K is Koppelberg compact, then P ( K ) = S ( K ). Problem 1 Is there a space K such that S 1 ( K ) � = S ( K )? Problem 2 Is there a space K such that S 1 ( K ) � = S ( K ) = P ( K )? Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Problems Theorem (Plebanek, PBN) If K is Koppelberg compact, then P ( K ) = S ( K ). Problem 1 Is there a space K such that S 1 ( K ) � = S ( K )? Problem 2 Is there a space K such that S 1 ( K ) � = S ( K ) = P ( K )? Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Asymptotic density Asymptotic density function We say that A ⊆ N has a density if the limit | A ∩ { 1 , 2 , . . . , n }| lim = d ( A ) n n →∞ exists. Density and weak ∗ convergence If every element of a Boolean algebra A ⊆ P ( N ) has a density, then for µ defined on the Stone space K of A by µ ( � A ) = d ( A ) for each A ∈ A we have δ 1 ( A ) + . . . + δ n ( A ) µ ( � A ) = lim . n n →∞ Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Asymptotic density Asymptotic density function We say that A ⊆ N has a density if the limit | A ∩ { 1 , 2 , . . . , n }| lim = d ( A ) n n →∞ exists. Density and weak ∗ convergence If every element of a Boolean algebra A ⊆ P ( N ) has a density, then for µ defined on the Stone space K of A by µ ( � A ) = d ( A ) for each A ∈ A we have δ 1 ( A ) + . . . + δ n ( A ) µ ( � A ) = lim . n n →∞ Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Asymptotic density Density and weak* convergence If every element of a Boolean algebra A ⊆ P ( N ) has a density, then for µ defined on the Stone space K of A by µ ( � A ) = d ( A ) for each A ∈ A we have δ 1 ( A ) + . . . + δ n ( A ) µ ( � A ) = lim . n n →∞ Corollary µ ∈ S 1 ( N ) ⊆ S 1 ( K ) . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Asymptotic density Density and weak* convergence If every element of a Boolean algebra A ⊆ P ( N ) has a density, then for µ defined on the Stone space K of A by µ ( � A ) = d ( A ) for each A ∈ A we have δ 1 ( A ) + . . . + δ n ( A ) µ ( � A ) = lim . n n →∞ Corollary µ ∈ S 1 ( N ) ⊆ S 1 ( K ) . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Asymptotic density Density and weak* convergence If every element of a Boolean algebra A ⊆ P ( N ) has a density, then for µ defined on the Stone space K of A by µ ( � A ) = d ( A ) for each A ∈ A we have δ 1 ( A ) + . . . + δ n ( A ) µ ( � A ) = lim . n n →∞ Corollary µ ∈ S 1 ( N ) ⊆ S 1 ( K ) . Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
Preliminaries Problems An example Limit of densities Relative density Fix a sequence ( B n ) n ∈ N of infinite and pairwise disjoint subsets of N such that � n B n = N . Let n ∈ N . Enumerate B n = { b 1 < b 2 < . . . } . For A ⊆ B n let d n ( A ) = d ( { i : b i ∈ A } ) . Limit of densities Let d ′ ( A ) = lim n →∞ d n ( A ) provided this limit exist. If each element A of a Boolean algebra A ⊆ P ( N ) is such that d ′ ( A ) exists, then µ ∈ P ( Stone ( A )) defined by µ ( � A ) = d ′ ( A ) is in S 2 ( N ) Piotr Borodulin–Nadzieja (Wroc� law) Sequential properties of measures
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