On the Lebesgue decomposition - Universiteit Leidenwebsites.math.leidenuniv.nl/positivity2013/presentations/cavaliere.pdf1joint work withPaolo de Lucia - Anna De Simone - Flavia Ventriglia,to

On the Lebesgue decomposition

for non-additive functions1

Paola CavaliereDepartment of Mathematics

University of Salerno

POSITIVITY VIIZaanen Centennial Conference

Leiden, July 25th, 2013

1joint work with Paolo de Lucia - Anna De Simone - Flavia Ventriglia, to appear.Paola Cavaliere (University of Salerno) On the Lebesgue decomposition Leiden, July 25th, 2013 1 / 13

The additive case on σ-algebras

Theorem (Lebesgue)

If A is a σ-algebra of sets and µ : A → R is a σ-additive measure,

then any σ-additive ϕ : A → R admits a unique decomposition

ϕ = ϕ1 + ϕ2

withI ϕ1 absolutely continuous with respect to µ, i.e. N (µ) ⊆ N (ϕ1);I ϕ2 singular with respect to µ, i.e.

there is some c ∈ N (ϕ2) such that c ′ ∈ N (µ).

Recall that the kernel of µ is defined by

N (µ) := {a ∈ A : µ(x) = 0 for every x ∈ A such that x ≤ a}.

Remark: µ : A → R σ-additive =⇒ N (µ) is a σ-ideal of A

Paola Cavaliere (University of Salerno) On the Lebesgue decomposition Leiden, July 25th, 2013 2 / 13


Theorem (Lebesgue)

If A is a σ-algebra of sets and µ : A → R is a σ-additive measure,then any σ-additive ϕ : A → R

admits a unique decomposition

ϕ = ϕ1 + ϕ2








Theorem (Lebesgue)

If A is a σ-algebra of sets and µ : A → R is a σ-additive measure,then any σ-additive ϕ : A → R admits a unique decomposition

ϕ = ϕ1 + ϕ2








Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2

withI ϕ1 absolutely continuous with respect to µ,

i.e. N (µ) ⊆ N (ϕ1);I ϕ2 singular with respect to µ, i.e.







Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2

withI ϕ1 absolutely continuous with respect to µ,

i.e. N (µ) ⊆ N (ϕ1);I ϕ2 singular with respect to µ, i.e.







Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2

withI ϕ1 absolutely continuous with respect to µ, i.e. N (µ) ⊆ N (ϕ1);

I ϕ2 singular with respect to µ, i.e.there is some c ∈ N (ϕ2) such that c ′ ∈ N (µ).






Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2

withI ϕ1 absolutely continuous with respect to µ, i.e. N (µ) ⊆ N (ϕ1);I ϕ2 singular with respect to µ,

i.e.there is some c ∈ N (ϕ2) such that c ′ ∈ N (µ).






Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2








Theorem (Lebesgue)


ϕ = ϕ1 + ϕ2







Theorem (Capek, 1981)

Let µ : A → S, where A is a σ-algebra and S a semigroup, suchthat

N (µ) is a σ-ideal of A.

Then for any ϕ : A → S′, with S′ a semigroup, fulfilling

1 ϕ(0) = 0 ,

2 N (µ) \ N (ϕ) satisfies the countable chain condition ,

there exists some c ∈ N (µ) such that

I ϕc′ : x ∈ A 7→ ϕ(x ∧ c ′) is µ-continuous;

I ϕc : x ∈ A 7→ ϕ(x ∧ c) is µ-singular.

Moreover, if ϕ is additive, then ϕ = ϕc′ + ϕc .

Recall

B ⊆ A is c.c.c.def⇐⇒

{disjoint collections in B \ {0} are

(at most) countable






1 ϕ(0) = 0 ,






Recall



(at most) countable






1 ϕ(0) = 0 ,






Recall



(at most) countable






1 ϕ(0) = 0 ,






Recall



(at most) countablePaola Cavaliere (University of Salerno) On the Lebesgue decomposition Leiden, July 25th, 2013 3 / 13





1 ϕ(0) = 0 ,






Recall








1 ϕ(0) = 0 ,






Recall








1 ϕ(0) = 0 ,






Recall




Notation

For B ⊆ A and c ∈ A

Bc := {x ∈ A : x ∧ c ∈ B} trace on c of B .


Let A be a σ-algebra, andM a σ-ideal of A.

For any N ⊆ A fulfilling

1 0 ∈ N,

2 M \N satisfies the countable chain condition,

then there exists some c ∈ M such that

M = Nc′ = Mc′ .

Hence,

M ⊆ Nc′ and Nc ⊥ M with c ∈ M and c ′ ∈ Nc

(Also, B. Riecan - T. Neubrunn (1997)).


Notation






1 0 ∈ N,



M = Nc′ = Mc′ .

Hence,




Notation






1 0 ∈ N,



M = Nc′ = Mc′ .

Hence,




Notation






1 0 ∈ N,



M = Nc′ = Mc′ .

Hence,




Notation






1 0 ∈ N,



M = Nc′ = Mc′ .

Hence,




Notation






1 0 ∈ N,



M = Nc′ = Mc′ .

Hence,




The additive case on orthomodular lattices

Let L = (L, 0, 1,∨,∧,′ ) be an OML,

i.e. an orthocomplemented bounded lattice in which

if a, b ∈ L, with a ≤ b, then b = a ∨ (b ∧ a′)

For a, b ∈ L, then

- a, b are orthogonal (a ⊥ b)def⇐⇒ a ≤ b′;

- a, b are perspective (a ∼ b)def⇐⇒

there is a c ∈ L such that

a ∧ c = b ∧ c = 0,

a ∨ c = b ∨ c = 1.

An ideal M of L is a p-idealdef⇐⇒ ∀a ∈ M: {x ∈ L : x ∼ a} ⊆ M .

An ideal M ⊆ L is a p–ideal ⇐⇒ ∀a ∈ M, x ∈ L: x ∧ (x ′ ∨ a) ∈ M.



Let L = (L, 0, 1,∨,∧,′ ) be an OML,







a ∧ c = b ∧ c = 0,

a ∨ c = b ∨ c = 1.





Let L = (L, 0, 1,∨,∧,′ ) be an OML,







a ∧ c = b ∧ c = 0,

a ∨ c = b ∨ c = 1.





Let L = (L, 0, 1,∨,∧,′ ) be an OML,







a ∧ c = b ∧ c = 0,

a ∨ c = b ∨ c = 1.





Let L = (L, 0, 1,∨,∧,′ ) be an OML,







a ∧ c = b ∧ c = 0,

a ∨ c = b ∨ c = 1.




Theorem (d’Andrea - de Lucia, 1991)

Let L be a σ-complete OML, and

M a σ-complete p-ideal of L.

For any N ⊆ L fulfilling

0 ∈ N, and M \N is c.c.c.,

then there exists some c ∈ M such that M = Nc′ = Mc′ .

Hence, M ⊆ Nc′ and Nc ⊥ M with c ∈ M and c ′ ∈ Nc


Let ϕ : L→ S, where L be a σ-complete OML, and S a semigroup,such that ϕ(0) = 0. Let M be a p-ideal of L.

IfM \ N (ϕ) satisfies the countable chain condition,

then there is a c ∈ M such that

I ϕc′ : x ∈ L 7→ ϕ(x ∧ c ′) is M-continuous i.e. M ⊂ N (ϕc′);

I ϕ⊥c′ : x ∈ L 7→ ϕ(x ∧ (x ′ ∨ c)) is M-singular i.e. M ⊥ N (ϕ⊥

c′).















c′).















c′).















c′).










Let ϕ : L→ S, where L be a σ-complete OML, and S a semigroup,such that ϕ(0) = 0.

Let M be a p-ideal of L.If

M \ N (ϕ) satisfies the countable chain condition,




c′).















c′).















c′).















c′).















c′).


Question: Let µ be a function defined on a OML.When does its kernel N (µ) be a p-ideal?

If µ : L→ S is finitely additive, with S a commutative semigroup,

N (µ) is a p–ideal ⇐⇒ µ is a p-function

1 µ is a p-functiondef⇔ ∀a ∈ L, b ∈ N (µ): µ(a ∨ b) = µ(a)

2 µ is null-additivedef⇔ ∀a ∈ L, b ∈ N (µ), s.t. a ⊥ b: µ(a∨ b) = µ(a).

Clearly,µ p-function =⇒ µ null-additive

I Definitions 1-2 still works for non-additive functions µ : L → S,with S = (S,U) a Hausdorff uniform space, on defining

N (µ) = {a ∈ L : µ(x) = µ(0) for every x ∈ L such that x ≤ a}.















































Hereafter, S = (S,U) is a Hausdorff uniform space.

Lemma (C., de Lucia, De Simone, Ventriglia (2013))

Let µ : L→ S be null-additive.

1 If N (µ) is a p–ideal, then µ is a p–function.

2 If µ is a p–function, then N (µ) is an ideal.

3 If µ is a p–function and for a, b ∈ L, with a ⊥ b,

µ(a ∨ b) = µ(a) =⇒ µ(b) = µ(0)

(i.e. µ is also weakly converse null–additive), then N (µ) is ap–ideal.

When L is σ-complete, one also has

For µ : L→ S, where L is σ-complete.,

µ is a σ–p-functiondef⇐⇒

{∀a ∈ L and orthogonal (dn)n∈N in N (µ)

µ(a) = µ(a ∨ (∨ndn)

)








µ(a ∨ b) = µ(a) =⇒ µ(b) = µ(0)






µ(a) = µ(a ∨ (∨ndn)

)








µ(a ∨ b) = µ(a) =⇒ µ(b) = µ(0)






µ(a) = µ(a ∨ (∨ndn)

)








µ(a ∨ b) = µ(a) =⇒ µ(b) = µ(0)






µ(a) = µ(a ∨ (∨ndn)

)








µ(a ∨ b) = µ(a) =⇒ µ(b) = µ(0)






µ(a) = µ(a ∨ (∨ndn)

)Paola Cavaliere (University of Salerno) On the Lebesgue decomposition Leiden, July 25th, 2013 8 / 13

In the same spirit of previous result on σ-complete OMLs


Let µ : L→ S be null-additive, with L σ-complete.

1 If N (µ) is a σ-complete p–ideal, then µ is a σ-p–function.

2 If µ is a σ-p–function,then N (µ) is a σ-complete ideal.

3 If µ is a σ-p–function and weakly converse null–additive, thenN (µ) is a σ-complete p–ideal.


s-outer functions

Let ϕ : L → S, with L an OML and S = (S,U) a Hausdorff uniformspace.

ϕ is s-outerdef⇐⇒

∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :(ϕ(a), ϕ(0)

)∈ V =⇒

(ϕ(a ∨ b), ϕ(b)

)∈ U.

Examples

Submeasures, measuroids, k-triangular functions and some decomposi-ble functions with respect to t-conorms [H. Weber, Saeki, Guselnikov,Pap, S. Weber...] are s-outer

Note

ϕ s-outer =⇒ ϕ null-additive


s-outer functions



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :(ϕ(a), ϕ(0)

)∈ V =⇒

(ϕ(a ∨ b), ϕ(b)

)∈ U.

Examples


Note



s-outer functions



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :(ϕ(a), ϕ(0)

)∈ V =⇒

(ϕ(a ∨ b), ϕ(b)

)∈ U.

Examples


Note



Let ϕ : L → S be s-outer. Then D = {ϕ1, ϕ2} is a decompositionof ϕ IF and only IF

1 ϕi : L→ (S,U) is s-outer and ϕi (0) = ϕ(0);

2 for any U ∈ U there exists some V ∈ U such that, for x ∈ L,

(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .

Theorem (C., de Lucia, De Simone, Ventriglia (2013))

Let ϕ : L→ S be s-outer -where L is a σ-complete OML, and S anHausdorff uniform space. Let M be a p-ideal of L.

If M \ N (ϕ) satisfies the countable chain condition,




c′).

Moreover, D = {ϕc′, ϕ⊥c′} is a decomposition of ϕ and such a

decomposition is unique.





(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .


Let ϕ : L→ S be s-outer -where L is a σ-complete OML, and S anHausdorff uniform space.

Let M be a p-ideal of L.If M \ N (ϕ) satisfies the countable chain condition,




c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).







(ϕ1(x), ϕ1(0)) ∈ V =⇒ (ϕ(x), ϕ2(x)) ∈ U ,

(ϕ2(x), ϕ2(0)) ∈ V =⇒ (ϕ(x), ϕ1(x)) ∈ U .







c′).




Idea of the proof.

1 D = {ϕc′, ϕ⊥c′} is a decomposition of ϕ. For this, ϕc′ and ϕ⊥

c′ mustbe s-outer. This is obtained by showing that for all a ⊥ b ∈ L

(i) ϕc′ (a ∨ b) = ϕ

((a ∧ c ′) ∨ (b ∧ c ′)

),

(ii) ϕ⊥c′

(a ∨ b) = ϕ((

a ∧ (a′ ∨ c))∨(b ∧ (b′ ∨ c)

)).

2 The proof is purely algebraic. It consists, in fact, in coupling thenull-additivity of ϕ with properties of p-ideals.


Let L be a σ-complete OML, and and µ : L→ So , where So is aHausdorff uniform space.

If µ is a σ–p-function that is weakly converse null–additive,thenany exhaustive s-outer ϕ : L→ S, where S is an Hausdorff uniformspace satisfying the first axiom of countability, admits a UNIQUEdecomposition D = {ϕ1, ϕ2} with

I ϕ1 µ–continuous and exhaustive;I ϕ2 µ–singular and exhaustive.


Idea of the proof.



(i) ϕc′ (a ∨ b) = ϕ

((a ∧ c ′) ∨ (b ∧ c ′)

),

(ii) ϕ⊥c′

(a ∨ b) = ϕ((

a ∧ (a′ ∨ c))∨(b ∧ (b′ ∨ c)

)).







Idea of the proof.



(i) ϕc′ (a ∨ b) = ϕ

((a ∧ c ′) ∨ (b ∧ c ′)

),

(ii) ϕ⊥c′

(a ∨ b) = ϕ((

a ∧ (a′ ∨ c))∨(b ∧ (b′ ∨ c)

)).




If µ is a σ–p-function that is weakly converse null–additive,

thenany exhaustive s-outer ϕ : L→ S, where S is an Hausdorff uniformspace satisfying the first axiom of countability, admits a UNIQUEdecomposition D = {ϕ1, ϕ2} with



Idea of the proof.



(i) ϕc′ (a ∨ b) = ϕ

((a ∧ c ′) ∨ (b ∧ c ′)

),

(ii) ϕ⊥c′

(a ∨ b) = ϕ((

a ∧ (a′ ∨ c))∨(b ∧ (b′ ∨ c)

)).




If µ is a σ–p-function that is weakly converse null–additive,thenany exhaustive s-outer ϕ : L→ S,

where S is an Hausdorff uniformspace satisfying the first axiom of countability, admits a UNIQUEdecomposition D = {ϕ1, ϕ2} with



Idea of the proof.



(i) ϕc′ (a ∨ b) = ϕ

((a ∧ c ′) ∨ (b ∧ c ′)

),

(ii) ϕ⊥c′

(a ∨ b) = ϕ((

a ∧ (a′ ∨ c))∨(b ∧ (b′ ∨ c)

)).








Let A be a σ-complete Boolean algebra, and µ : A → So , whereSo is a Hausdorff uniform space.

If µ is order–continuous and either null–additive or quasi–triangular,then any exhaustive s-outer ϕ : L→ S , where S is an Hausdorffuniform space satisfying the first axiom of countability, admits a uniquedecomposition D = {ϕ1, ϕ2} with


ϕ quasi-triangulardef⇔

∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :

ϕ(a) ∈ V , ϕ(b) ∈ V =⇒ ϕ(a ∨ b) ∈ U;

ϕ(a) ∈ V , ϕ(a ∨ b) ∈ V =⇒ ϕ(b) ∈ U.



Let A be a σ-complete Boolean algebra, and µ : A → So , whereSo is a Hausdorff uniform space.If µ is order–continuous and either null–additive or quasi–triangular

,then any exhaustive s-outer ϕ : L→ S , where S is an Hausdorffuniform space satisfying the first axiom of countability, admits a uniquedecomposition D = {ϕ1, ϕ2} with



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :

ϕ(a) ∈ V , ϕ(b) ∈ V =⇒ ϕ(a ∨ b) ∈ U;

ϕ(a) ∈ V , ϕ(a ∨ b) ∈ V =⇒ ϕ(b) ∈ U.



Let A be a σ-complete Boolean algebra, and µ : A → So , whereSo is a Hausdorff uniform space.If µ is order–continuous and either null–additive or quasi–triangular

,then any exhaustive s-outer ϕ : L→ S , where S is an Hausdorffuniform space satisfying the first axiom of countability, admits a uniquedecomposition D = {ϕ1, ϕ2} with



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :

ϕ(a) ∈ V , ϕ(b) ∈ V =⇒ ϕ(a ∨ b) ∈ U;

ϕ(a) ∈ V , ϕ(a ∨ b) ∈ V =⇒ ϕ(b) ∈ U.



Let A be a σ-complete Boolean algebra, and µ : A → So , whereSo is a Hausdorff uniform space.If µ is order–continuous and either null–additive or quasi–triangular,

then any exhaustive s-outer ϕ : L→ S , where S is an Hausdorffuniform space satisfying the first axiom of countability, admits a uniquedecomposition D = {ϕ1, ϕ2} with



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :

ϕ(a) ∈ V , ϕ(b) ∈ V =⇒ ϕ(a ∨ b) ∈ U;

ϕ(a) ∈ V , ϕ(a ∨ b) ∈ V =⇒ ϕ(b) ∈ U.



Let A be a σ-complete Boolean algebra, and µ : A → So , whereSo is a Hausdorff uniform space.If µ is order–continuous and either null–additive or quasi–triangular,

then any exhaustive s-outer ϕ : L→ S , where S is an Hausdorffuniform space satisfying the first axiom of countability, admits a uniquedecomposition D = {ϕ1, ϕ2} with



∀U ∈ U ∃V = V (U) ∈ U s.t.

∀a, b ∈ L, a ⊥ b :

ϕ(a) ∈ V , ϕ(b) ∈ V =⇒ ϕ(a ∨ b) ∈ U;

ϕ(a) ∈ V , ϕ(a ∨ b) ∈ V =⇒ ϕ(b) ∈ U.


Documents

On the Lebesgue decomposition - Universiteit Leidenwebsites.math.leidenuniv.nl/positivity2013/presentations/cavaliere.pdf1joint work withPaolo de Lucia - Anna De Simone - Flavia Ventriglia,to