A pointfree account of Carath eodorys Extension Theorem s Jakl a - PowerPoint PPT Presentation

A pointfree account of Carath´ eodory’s Extension Theorem s Jakl a Tom´ aˇ Workshop on Algebra, Logic and Topology in Coimbra 27 September 2018 a The research discussed has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.670624)

Classical Carath´ eodory’s Extension Theorem Theorem A measure m : B → [0 , 1] on a Boolean algebra B ⊆ P ( X ) uniquely extends to a countably additive measure on σ ( B ) . Minimal σ -algebra contaning B 1

Classical Carath´ eodory’s Extension Theorem Theorem A measure m : B → [0 , 1] on a Boolean algebra B ⊆ P ( X ) uniquely extends to a countably additive measure on σ ( B ) . Proof. 1. Extend m to a countably additive Minimal σ -algebra function B contaning B m µ ( U ) = sup { m ( B ) | B ∈ B , B ⊆ U } µ 2. Extend µ to an outer measure [0 , 1] τ B µ ∗ ( M ) = inf { µ ( U ) | U ∈ τ B , M ⊆ U } µ ∗ 3. µ ∗ is a measure on measurable subsets P ( X ) H ⊆ P ( X ). Restrict µ ∗ to σ ( B ) ⊆ H . 1

Extension theorem by Igor Kˇ r´ ıˇ z and Aleˇ s Pultr Abstract σ -algebra is a Boolean algebra which has countable joins. Abstract finitely (resp. countably) additive measure m : B → [0 , 1] satisfies 1. m (0 B ) = 0, m (1 B ) = 1, 2. m ( a ∨ b ) + m ( a ∧ b ) = m ( a ) + m ( b ) 3. (resp. � ∞ i =0 m ( a i ) = m ( � ∞ i =0 a i ) if a i ’s are pairwise disjoint) 2

Extension theorem by Igor Kˇ r´ ıˇ z and Aleˇ s Pultr Abstract σ -algebra is a Boolean algebra which has countable joins. Abstract finitely (resp. countably) additive measure m : B → [0 , 1] satisfies 1. m (0 B ) = 0, m (1 B ) = 1, 2. m ( a ∨ b ) + m ( a ∧ b ) = m ( a ) + m ( b ) 3. (resp. � ∞ i =0 m ( a i ) = m ( � ∞ i =0 a i ) if a i ’s are pairwise disjoint) Theorem (Kˇ r´ ıˇ z, Pultr 2010) B Every finitely additive m : B → [0 , 1] m uniquely extends to a countably additive measure µ : σ Alg � B � → [0 , 1] such that σ Alg � B � [0 , 1] µ Enlarges the space. On the other hand, useful for integration over infinite-dimensional spaces! 2

What instead of P ( X ) ? Finitely additive m : B → [0 , 1] extends B to a valuation µ : Idl( B ) → [0 , 1], m µ ( I ) = sup { m ( a ) : a ∈ I } µ Idl( B ) [0 , 1] ??? 3

What instead of P ( X ) ? Finitely additive m : B → [0 , 1] extends B to a valuation µ : Idl( B ) → [0 , 1], i.e. 1. µ is a finitely additive measure m 2. For a directed A ⊆ ↑ Idl( B ): µ ( I ) = sup { m ( a ) : a ∈ I } µ Idl( B ) [0 , 1] � ↑ A ) sup µ ( I ) = µ ( I ∈ A ??? 3

What instead of P ( X ) ? Finitely additive m : B → [0 , 1] extends B to a valuation µ : Idl( B ) → [0 , 1], i.e. 1. µ is a finitely additive measure m 2. For a directed A ⊆ ↑ Idl( B ): µ ( I ) = sup { m ( a ) : a ∈ I } µ Idl( B ) [0 , 1] � ↑ A ) sup µ ( I ) = µ ( I ∈ A ??? We need a complete Boolean algebra which • embeds Idl( B ), and • has the same (frame-theoretic) points as B has. 3

What instead of P ( X ) ? Finitely additive m : B → [0 , 1] extends B to a valuation µ : Idl( B ) → [0 , 1], i.e. 1. µ is a finitely additive measure m 2. For a directed A ⊆ ↑ Idl( B ): µ ( I ) = sup { m ( a ) : a ∈ I } µ Idl( B ) [0 , 1] � ↑ A ) sup µ ( I ) = µ ( I ∈ A ??? We need a complete Boolean algebra which Idl( B ) is a frame! • embeds Idl( B ), and • has the same (frame-theoretic) a ∧ � i b i = � i ( a ∧ b i ) points as B has. e.g. O ( X , τ ) = τ 3

Frame Theory intermezzo: Sublocales A subspace M ⊆ X introduces a frame congruence ∼ M on O ( X ): U ∼ M V iff U ∩ M = V ∩ M 4

Frame Theory intermezzo: Sublocales A subspace M ⊆ X introduces a frame congruence ∼ M on O ( X ): U ∼ M V iff U ∩ M = V ∩ M Congruences are equivalently represented as sublocales S ⊆ L � A ∈ S 1. ∀ A ⊆ S , 2. ∀ x ∈ L , s ∈ S , x → s ∈ S 4

Frame Theory intermezzo: Sublocales A subspace M ⊆ X introduces a frame congruence ∼ M on O ( X ): U ∼ M V iff U ∩ M = V ∩ M Congruences are equivalently represented as sublocales S ⊆ L � A ∈ S 1. ∀ A ⊆ S , 2. ∀ x ∈ L , s ∈ S , x → s ∈ S The mapping “congruences �→ sublocales”: ∼ ⊆ L × L �− → { largest elements of ∼ -equivalence classes } Every subspace of X introduces a sublocale of O ( X ) but not vice versa! 4

The complete lattice (coframe) of sublocales S ( L ) = { S ⊆ L | S is a sublocale } , ordered by ⊆ . Joins and meet easy to compute! 5

The complete lattice (coframe) of sublocales S ( L ) = { S ⊆ L | S is a sublocale } , ordered by ⊆ . Joins and meet easy to compute! Open and closed sublocales ( a ∈ L ): o ( a ) = { a → x | x ∈ L } and c ( a ) = ↑ a They are complemented in S ( L ). � i o ( a i ) = o ( � i a i ), c ( a ) ∨ c ( b ) = c ( a ∧ b ), ... (as expected) 5

The complete lattice (coframe) of sublocales S ( L ) = { S ⊆ L | S is a sublocale } , ordered by ⊆ . Joins and meet easy to compute! Open and closed sublocales ( a ∈ L ): o ( a ) = { a → x | x ∈ L } and c ( a ) = ↑ a They are complemented in S ( L ). � i o ( a i ) = o ( � i a i ), c ( a ) ∨ c ( b ) = c ( a ∧ b ), ... (as expected) Join-sublattice S c ( L ) ⊆ S ( L ) � � the set of sublocales obtained as S c ( L ) = joins of closed sublocales Always a frame! 5

Theorem (Picado, Pultr, Tozzi 2016) If L is subfit then S c ( L ) is a complete Boolean algebra and a ∈ L �− → o ( a ) ∈ S c ( L ) is an injective frame homomorphisms L ֒ → S c ( L ) . 6

Theorem (Picado, Pultr, Tozzi 2016) If L is subfit then S c ( L ) is a complete Boolean algebra and a ∈ L �− → o ( a ) ∈ S c ( L ) is an injective frame homomorphisms L ֒ → S c ( L ) . Moreover • If X is a T 1 space, then S c ( O ( X )) ∼ = P ( X ). • In case of X = spec( B ), we have O ( X ) ∼ = Idl( B ) and so S c (Idl( B )) ∼ = P ( X ) . 6

Theorem (Picado, Pultr, Tozzi 2016) If L is subfit then S c ( L ) is a complete Boolean algebra and a ∈ L �− → o ( a ) ∈ S c ( L ) is an injective frame homomorphisms L ֒ → S c ( L ) . Moreover • If X is a T 1 space, then S c ( O ( X )) ∼ = P ( X ). • In case of X = spec( B ), we have O ( X ) ∼ = Idl( B ) and so S c (Idl( B )) ∼ = P ( X ) . • = ⇒ instead of P ( X ) take S c (Idl( B )) 6

Putting it together B Valuation µ : Idl( B ) → [0 , 1] extends to an outer measure µ ∗ : S c (Idl( B )) → [0 , 1], m µ Idl( B ) [0 , 1] µ ∗ ( x ) = inf { µ ( i ) | i ∈ Idl( B ) , x ≤ i } µ ∗ S c (Idl( B )) 7

Putting it together B Valuation µ : Idl( B ) → [0 , 1] extends to an outer measure µ ∗ : S c (Idl( B )) → [0 , 1], i.e. m 1. µ ∗ is monotone µ Idl( B ) [0 , 1] 2. µ ∗ ( x ∨ y ) + µ ∗ ( x ∧ y ) ≤ µ ∗ ( a ) + µ ∗ ( b ) µ ∗ ( x ) = inf { µ ( i ) | i ∈ Idl( B ) , x ≤ i } i =0 ⊆ ↑ S c (Idl( B )): 3. For a directed ( x i ) ∞ µ ∗ S c (Idl( B )) µ ∗ ( x i ) = µ ∗ ( � ↑ sup x i ) i i 7

Putting it together B Valuation µ : Idl( B ) → [0 , 1] extends to an outer measure µ ∗ : S c (Idl( B )) → [0 , 1], i.e. m 1. µ ∗ is monotone µ Idl( B ) [0 , 1] 2. µ ∗ ( x ∨ y ) + µ ∗ ( x ∧ y ) ≤ µ ∗ ( a ) + µ ∗ ( b ) µ ∗ ( x ) = inf { µ ( i ) | i ∈ Idl( B ) , x ≤ i } i =0 ⊆ ↑ S c (Idl( B )): 3. For a directed ( x i ) ∞ µ ∗ S c (Idl( B )) µ ∗ ( x i ) = µ ∗ ( � ↑ sup x i ) i i Furthermore H = { x ∈ S c (Idl( B )) | µ ∗ ( x ) + µ ∗ ( ¬ x ) ≤ 1 } is a σ -algebra (containing σ S ( B )) and so µ ∗ ↾ H is a measure. 7

Pointfree Carath´ eodory’s Extension Theorem Theorem A finitely additive measure m : B → [0 , 1] uniquely extends to a countably additive measure on σ S ( B ) ⊆ S c (Idl( B )) . 8

Pointfree Carath´ eodory’s Extension Theorem Theorem A finitely additive measure m : B → [0 , 1] uniquely extends to a countably additive measure on σ S ( B ) ⊆ S c (Idl( B )) . Corollary There are bijective correspondences between • finitely additive measures B → [0 , 1] • regular countably additive measures σ S ( B ) → [0 , 1] • regular valuations σ S (Idl( B )) → [0 , 1] 8

Comparison with the classical result For a Boolean algebra B ⊆ P ( X ), it might happen that � i B i ∈ B for some infinite { B i } i ⊆ B . However, in the Stone space spec( B ) (i.e. in the “sobrification”) � ◦ �� � � i � B i � � = � i B i � = i � B i � where � B � = {U | B ∈ U} . 9

Comparison with the classical result For a Boolean algebra B ⊆ P ( X ), it might happen that � i B i ∈ B for some infinite { B i } i ⊆ B . However, in the Stone space spec( B ) (i.e. in the “sobrification”) � ◦ �� � � i � B i � � = � i B i � = i � B i � where � B � = {U | B ∈ U} . = ⇒ We don’t need the extra assumption for m : B → [0 , 1]: For any pairwise disjoint { B i } ∞ � i =0 ⊆ B such that i B i ∈ B ∞ � � m ( i B i ) = m ( B i ) i =0 9

The continuous map U : ( X , P ( X )) → (spec( B ) , P (spec( B ))) U : x �− → { B ∈ B | x ∈ B } 10

The continuous map U : ( X , P ( X )) → (spec( B ) , P (spec( B ))) U : x �− → { B ∈ B | x ∈ B } introduces a frame homomorphism h : P (spec( B )) → P ( X ) h : M �→ { x | U ( x ) ∈ M } 10