An Effect- Theoretic Account of Lebesgue Integration Bart Jacobs - - PowerPoint PPT Presentation

An Effect- Theoretic Account of Lebesgue Integration

Bart Jacobs bart@cs.ru.nl Bram Westerbaan awesterb@cs.ru.nl

Radboud University Nijmegen

June 23, 2015

Some locals

Our usual business: categorical program semantics

• Predicate

transformers

• p
• ⊤
• State

transformers

• Programs
• Pred
• Stat

Our usual business: semantics of quantum programs

• Effect algebras
• p
• ⊤
• Convex sets
• Von Neumann

algebras

• p

Pred

• Stat

Our usual business: effectus theory

• Effect algebras
• p
• ⊤
• Convex sets
• Effectus
• Pred
• Stat

Some related work

Some related work

• 1. method of exhaustion by Eudoxos, ∼390, BC

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC

Method of exhaustion

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .
• 3. formalised by Riemann, 1854

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .
• 3. formalised by Riemann, 1854
• 4. completed by Lebesgue, 1902

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .
• 3. formalised by Riemann, 1854
• 4. completed by Lebesgue, 1902
• 5. generalised by Daniell in 1918, Bochner in 1933, Haar in

1940, Pettis around 1943, Stone in 1948, . . .

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .
• 3. formalised by Riemann, 1854
• 4. completed by Lebesgue, 1902
• 5. generalised by Daniell in 1918, Bochner in 1933, Haar in

1940, Pettis around 1943, Stone in 1948, . . .

• 6. we present another generalisation* based on effect algebras

Short history of integration

• 1. method of exhaustion by Eudoxos, ∼390, BC
• 2. integration of functions Newton, ∼1665, . . .
• 3. formalised by Riemann, 1854
• 4. completed by Lebesgue, 1902
• 5. generalised by Daniell in 1918, Bochner in 1933, Haar in

1940, Pettis around 1943, Stone in 1948, . . .

• 6. we present another generalisation* based on effect algebras

* of integration of [0, 1]-valued functions with respect to probability measures (≈ [0, 1]-valued measures)

But why yet another !?

But why yet another !?

The theory of integration, because of its central rˆ

• le in mathematical analysis and

geometry, continues to afford opportunities for serious investigation. — M.H. Stone, 1948

But why yet another !?

The theory of integration, because of its central rˆ

• le in mathematical analysis and

geometry, continues to afford opportunities for serious investigation. — M.H. Stone, 1948

Universal property

(measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

Universal property

(measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Effect algebras

An effect algebra is a set E with 0, 1, (−)⊥, and partial Examples:

• 1. [0, 1]

a b = a + b if a + b ≤ 1

• 2. ℘(X)

A B = A ∪ B if A ∩ B = ∅

Effect algebras

An effect algebra is a set E with 0, 1, (−)⊥, and partial with

• 1. a b = b a
• 2. a (b c) = (a b) c
• 3. a 0 = a
• 4. a a⊥ = 1

Examples:

• 1. [0, 1]

a b = a + b if a + b ≤ 1

• 2. ℘(X)

A B = A ∪ B if A ∩ B = ∅

Effect algebras

An effect algebra is a set E with 0, 1, (−)⊥, and partial with

• 1. a b = b a
• 2. a (b c) = (a b) c
• 3. a 0 = a
• 4. a a⊥ = 1
• 5. a b = 0

= ⇒ a = b = 0

• 6. a b = a c

= ⇒ b = c Examples:

• 1. [0, 1]

a b = a + b if a + b ≤ 1

• 2. ℘(X)

A B = A ∪ B if A ∩ B = ∅

Effect algebras

An effect algebra is a set E with 0, 1, (−)⊥, and partial with

• 1. a b = b a
• 2. a (b c) = (a b) c
• 3. a 0 = a
• 4. a a⊥ = 1
• 5. a b = 0

= ⇒ a = b = 0

• 6. a b = a c

= ⇒ b = c Examples:

• 1. [0, 1]

a b = a + b if a + b ≤ 1

• 2. ℘(X)

A B = A ∪ B if A ∩ B = ∅

• 3. Ef (H)

A B = A + B if A + B ≤ I

Universal property

(measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

(measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

ω-complete effect algebras

ω-complete effect algebras

Let E be an effect algebra. a ≤ b ⇐ ⇒ ∃d a d = b

ω-complete effect algebras

Let E be an effect algebra. a ≤ b ⇐ ⇒ ∃d a d = b The effect algebra E is ω-complete if each chain a1 ≤ a2 ≤ · · · has a supremum,

n an.

ω-complete effect algebras

Let E be an effect algebra. a ≤ b ⇐ ⇒ ∃d a d = b The effect algebra E is ω-complete if each chain a1 ≤ a2 ≤ · · · has a supremum,

n an.

Examples:

• 1. [0, 1]
• 2. ℘(X)
• n An =

n An

• 3. Ef (H)

ω-complete effect algebras

Let E be an effect algebra. a ≤ b ⇐ ⇒ ∃d a d = b The effect algebra E is ω-complete if each chain a1 ≤ a2 ≤ · · · has a supremum,

n an.

Examples:

• 1. [0, 1]
• 2. ℘(X)
• n An =

n An

• 3. Ef (H)
• 4. σ-algebra on X

= sub-(ω-complete EA) of ℘(X) !

Universal property

(measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. (measurable subsets)

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Measurable functions

Let ΣX be a σ-algebra on a set X A map f : X → [0, 1] is measurable if f −1([a, b]) ∈ ΣX for all a ≤ b in [0, 1] Meas(X, [0, 1]) = { f : X → [0, 1]: f is measurable }

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• (measurable functions)

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Homomorphisms of (ω-complete) effect algebras

f : F → E is a homomorphism of effect algebras if

• 1. f (0) = 0

f (1) = 1 f (a⊥) = f (a)⊥

• 2. if a b is defined, then

f (a b) = f (a) f (b)

Homomorphisms of (ω-complete) effect algebras

f : F → E is a homomorphism of effect algebras if

• 1. f (0) = 0

f (1) = 1 f (a⊥) = f (a)⊥

• 2. if a b is defined, then

f (a b) = f (a) f (b) f is a homomorphism of ω-complete effect algebras if 3.

n f (an) = f ( n an)

for a1 ≤ a2 ≤ · · · in F

Homomorphisms of (ω-complete) effect algebras

f : F → E is a homomorphism of effect algebras if

• 1. f (0) = 0

f (1) = 1 f (a⊥) = f (a)⊥

• 2. if a b is defined, then

f (a b) = f (a) f (b) f is a homomorphism of ω-complete effect algebras if 3.

n f (an) = f ( n an)

for a1 ≤ a2 ≤ · · · in F Examples:

• 1. 1(−) : ΣX −

→ Meas(X, [0, 1])

Homomorphisms of (ω-complete) effect algebras

f : F → E is a homomorphism of effect algebras if

• 1. f (0) = 0

f (1) = 1 f (a⊥) = f (a)⊥

• 2. if a b is defined, then

f (a b) = f (a) f (b) f is a homomorphism of ω-complete effect algebras if 3.

n f (an) = f ( n an)

for a1 ≤ a2 ≤ · · · in F Examples:

• 1. 1(−) : ΣX −

→ Meas(X, [0, 1])

• 2. homomorphisms of ω-complete EA

µ: ΣX → [0, 1] = probability measures on X (!)

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

key observation: both µ and

• (−)dµ are

homomorphisms of ω-complete effect algebras

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of (...?...)

• (−)dµ

such that

• 1Adµ = µ(A).

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of (...?...)

• (−)dµ

such that

• 1Adµ = µ(A).

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of ω-complete effect modules

• (−)dµ

such that

• 1Adµ = µ(A).

Effect modules

An effect module is an effect algebra E with scalar multiplication λ · x (λ ∈ [0, 1], x ∈ E)

Effect modules

An effect module is an effect algebra E with scalar multiplication λ · x (λ ∈ [0, 1], x ∈ E) such that

• 1. 1 · a = a
• 2. λ · (µ · a) = (λ · µ) · a
• 3. λ · (−) preserves and 0
• 4. (−) · a preserves and 0

Effect modules

An effect module is an effect algebra E with scalar multiplication λ · x (λ ∈ [0, 1], x ∈ E) such that

• 1. 1 · a = a
• 2. λ · (µ · a) = (λ · µ) · a
• 3. λ · (−) preserves and 0
• 4. (−) · a preserves and 0

Examples:

• 1. [0, 1], Ef (H), Meas(X, [0, 1]) are
• 2. ℘(X) is not

Effect modules

An effect module is an effect algebra E with scalar multiplication λ · x (λ ∈ [0, 1], x ∈ E) such that

• 1. 1 · a = a
• 2. λ · (µ · a) = (λ · µ) · a
• 3. λ · (−) preserves and 0
• 4. (−) · a preserves and 0

Examples:

• 1. [0, 1], Ef (H), Meas(X, [0, 1]) are
• 2. ℘(X) is not

A homomorphism of effect modules is what you expect

Universal property

Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• [0, 1]

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of ω-complete effect modules

• (−)dµ

such that

• 1Adµ = µ(A).

Universal property

Let E be an ω-complete effect module. Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• E

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of ω-complete effect modules

• (−)dµ

such that

• 1Adµ = µ(A).

Universal property

Let E be an ω-complete effect module. Let ΣX be a σ-algebra on a set X. ΣX

A→1A

• µ
• Meas(X, [0, 1])

f →

• f dµ
• E

For every homomorphism of ω-complete effect algebras µ there is a unique hom. of ω-complete effect modules

• (−)dµ

such that

• 1Adµ = µ(A).

Conclusion: Meas(X, [0, 1]) is the free ω-complete effect module

• ver ΣX via A → 1A.

Sketch of the proof

ΣX

A → 1A

• µ

E

Meas(X, [0, 1])

Sketch of the proof

ΣX

µ

• A → 1A
• (step functions)
• E

Meas(X, [0, 1])

Sketch of the proof

ΣX

µ

• A → 1A
• (step functions)

λn1An → λnµ(An)

• E

Meas(X, [0, 1])

Sketch of the proof

ΣX

µ

• A → 1A
• (step functions)

λn1An → λnµ(An)

• E

Meas(X, [0, 1])

fn → fn

Example: formulation of the Spectral Theorem

Let H be a Hilbert space.

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H).

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible}

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible} Spectral theorem: there is a unique homomorphism of ω-complete effect algebras φ: ΣσA − → Ef (H) such that

• 1. A =
• id dφ
• 2. φ(S) is a projection for all S ∈ ΣσA

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible} Spectral theorem: there is a unique homomorphism of ω-complete effect algebras φ: ΣσA − → Ef (H) such that

• 1. A =
• id dφ
• 2. φ(S) is a projection for all S ∈ ΣσA
• 3. if φ(G) = 0 for an open subset of σA, then G = ∅

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible} Spectral theorem: there is a unique homomorphism of ω-complete effect algebras φ: ΣσA − → Ef (H) such that

• 1. A =
• id dφ
• 2. φ(S) is a projection for all S ∈ ΣσA
• 3. if φ(G) = 0 for an open subset of σA, then G = ∅

Moreover, we have 4.

• f dφ ·
• gdφ =
• f · g dφ

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible} Spectral theorem: there is a unique homomorphism of ω-complete effect algebras φ: ΣσA − → Ef (H) such that

• 1. A =
• id dφ
• 2. φ(S) is a projection for all S ∈ ΣσA
• 3. if φ(G) = 0 for an open subset of σA, then G = ∅

Moreover, we have 4.

• f dφ ·
• gdφ =
• f · g dφ
• 5. B commutes with A iff B commutes with all φ(S)

Example: formulation of the Spectral Theorem

Let H be a Hilbert space. Let A be an effect on H — A ∈ Ef (H). Let σ(A) = {λ ∈ C: A − λ is not invertible} Spectral theorem: there is a unique homomorphism of ω-complete effect algebras φ: ΣσA − → Ef (H) such that

• 1. A =
• id dφ
• 2. φ(S) is a projection for all S ∈ ΣσA
• 3. if φ(G) = 0 for an open subset of σA, then G = ∅

Moreover, we have 4.

• f dφ ·
• gdφ =
• f · g dφ
• 5. B commutes with A iff B commutes with all φ(S)

Motto: effects behave somewhat like measurable functions; the integral

• (−)dφ: Meas(X, [0, 1]) → Ef (H) translates.

Recap and outlook

You have seen:

• 1. Lebesgue integration and effect algebras.
• 2. A universal property of the extension of measure to integral.

Agenda:

• 1. Fubini’s Theorem
• 2. Carath´

eodory’s Extension Theorem

• 3. Gleason’s Theorem