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Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Elementary numerosities and measures

Emanuele Bottazzi, University of Trento Naples - Konstanz Model Theory Days 2013

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Finitely additive measures

Definition A finitely additive measure is a triple (Ω, A, µ) where: The space Ω is a non-empty set; A is a ring of sets over Ω, i.e. a non-empty family of subsets of Ω satisfying the conditions: A, B ∈ A ⇒ A ∪ B, A ∩ B, A \ B ∈ A; µ : A → [0, +∞]R is an additive function, i.e. µ(A ∪ B) = µ(A) + µ(B) whenever A, B ∈ A are disjoint. We also assume that µ(∅) = 0.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Finitely additive measures

Definition A finitely additive measure is a triple (Ω, A, µ) where: The space Ω is a non-empty set; A is a ring of sets over Ω, i.e. a non-empty family of subsets of Ω satisfying the conditions: A, B ∈ A ⇒ A ∪ B, A ∩ B, A \ B ∈ A; µ : A → [0, +∞]R is an additive function, i.e. µ(A ∪ B) = µ(A) + µ(B) whenever A, B ∈ A are disjoint. We also assume that µ(∅) = 0. A measure (Ω, A, µ) is called non-atomic when all finite sets in A have measure zero.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Superreal Fields

Definition Let F be an ordered field that contains N. A number ξ ∈ F is called infinitesimal if |ξ| < 1/n for all n ∈ N. It is called infinite if |ξ| > n for all n ∈ N.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Superreal Fields

Definition Let F be an ordered field that contains N. A number ξ ∈ F is called infinitesimal if |ξ| < 1/n for all n ∈ N. It is called infinite if |ξ| > n for all n ∈ N. Proposition There exist ordered fields that properly extend R. Such fields are non-archimedean, i.e. they contain infinite and (nonzero) infinitesimal numbers.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Superreal Fields

Definition Let F be an ordered field that contains N. A number ξ ∈ F is called infinitesimal if |ξ| < 1/n for all n ∈ N. It is called infinite if |ξ| > n for all n ∈ N. Proposition There exist ordered fields that properly extend R. Such fields are non-archimedean, i.e. they contain infinite and (nonzero) infinitesimal numbers. Definition A superreal field is an ordered field F that properly extends R.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

The Standard Part

Proposition Let F be a superreal field. Every finite number ξ ∈ F can be represented in a unique way by ξ = sh(ξ) + ǫ where sh(ξ) ∈ R and ǫ = ξ − sh(ξ) is infinitesimal.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

The Standard Part

Proposition Let F be a superreal field. Every finite number ξ ∈ F can be represented in a unique way by ξ = sh(ξ) + ǫ where sh(ξ) ∈ R and ǫ = ξ − sh(ξ) is infinitesimal. We also define sh(ξ) = +∞ whenever ξ is positive infinite, and sh(ξ) = −∞ whenever ξ is negative infinite.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Elementary numerosities

Definition An elementary numerosity on a set Ω is a function n : P(Ω) → [0, +∞)F defined for all subsets of Ω, taking values into the non-negative part of a superreal field F, and satifying the conditions: n({x}) = 1 for every point x ∈ Ω ; n(A ∪ B) = n(A) + n(B) whenever A and B are disjoint.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Numerosity measures

Proposition Let n : P(Ω) → [0, +∞)F be an elementary numerosity, and for every β > 0 in F define the function nβ : P(Ω) → [0, +∞]R by posing nβ(A) = sh n(A) β

• .

Then nβ is a finitely additive measure defined for all subsets of Ω. Moreover, nβ is non-atomic if and only if β is an infinite number.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

The main result

Theorem Let (Ω, A, µ) be a non-atomic finitely additive measure. Then there exist a non-archimedean field F ⊇ R ; an elementary numerosity n : P(Ω) → [0, +∞)F ; such that for every positive number of the form β = n(X)

µ(X) one

has µ(A) = nβ(A) for all A ∈ A.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

The main result

Theorem Let (Ω, A, µ) be a non-atomic finitely additive measure. Then there exist a non-archimedean field F ⊇ R ; an elementary numerosity n : P(Ω) → [0, +∞)F ; such that for every positive number of the form β = n(X)

µ(X) one

has µ(A) = nβ(A) for all A ∈ A. Moreover, if B ⊆ A is a subring whose non-empty sets have all positive measure, then we can also ask that n(B) = n(B′) for all B, B′ ∈ B such that µ(B) = µ(B′).

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: Lebesgue measure

Corollary Let (R, L, µL) be the Lebesgue measure over R. There exists an elementary numerosity n : P(R) → F such that:

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: Lebesgue measure

Corollary Let (R, L, µL) be the Lebesgue measure over R. There exists an elementary numerosity n : P(R) → F such that: sh

• n(A)

n([0,1))

• = µL(A) for all A ∈ L.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: Lebesgue measure

Corollary Let (R, L, µL) be the Lebesgue measure over R. There exists an elementary numerosity n : P(R) → F such that: sh

• n(A)

n([0,1))

• = µL(A) for all A ∈ L.

sh

• n(A)

n([0,1))

• ≤ µL(A) for all A ⊆ R.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: Lebesgue measure

Corollary Let (R, L, µL) be the Lebesgue measure over R. There exists an elementary numerosity n : P(R) → F such that: sh

• n(A)

n([0,1))

• = µL(A) for all A ∈ L.

sh

• n(A)

n([0,1))

• ≤ µL(A) for all A ⊆ R.

n([x, x + a)) = n([y, y + a)) for all x, y ∈ R and for all a > 0.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: Lebesgue measure

Corollary Let (R, L, µL) be the Lebesgue measure over R. There exists an elementary numerosity n : P(R) → F such that: sh

• n(A)

n([0,1))

• = µL(A) for all A ∈ L.

sh

• n(A)

n([0,1))

• ≤ µL(A) for all A ⊆ R.

n([x, x + a)) = n([y, y + a)) for all x, y ∈ R and for all a > 0. n([x, x + a)) = a · n([0, 1)) for all rational numbers a > 0.

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: the fair coin measure

Corollary Let (2N, A, µ) be the fair coin measure. There exists an elementary numerosity n : P(2N) → F such that the function P(A) = n(A)/n(2N) satisfies the conditions:

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: the fair coin measure

Corollary Let (2N, A, µ) be the fair coin measure. There exists an elementary numerosity n : P(2N) → F such that the function P(A) = n(A)/n(2N) satisfies the conditions:

1 sh(P(A)) = µ(A) for all A ∈ A

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: the fair coin measure

Corollary Let (2N, A, µ) be the fair coin measure. There exists an elementary numerosity n : P(2N) → F such that the function P(A) = n(A)/n(2N) satisfies the conditions:

1 sh(P(A)) = µ(A) for all A ∈ A 2 P agrees with µ over all cylindrical sets:

P

• C (i1,...,in)

(t1,...,tn)

• = µ
• C (i1,...,in)

(t1,...,tn)

• = 2−n

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

An application: the fair coin measure

Corollary Let (2N, A, µ) be the fair coin measure. There exists an elementary numerosity n : P(2N) → F such that the function P(A) = n(A)/n(2N) satisfies the conditions:

1 sh(P(A)) = µ(A) for all A ∈ A 2 P agrees with µ over all cylindrical sets:

P

• C (i1,...,in)

(t1,...,tn)

• = µ
• C (i1,...,in)

(t1,...,tn)

• = 2−n

3 if F ⊂ 2N is finite, then for all A ⊆ 2N,

P(A|F) = |A ∩ F| |F|

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Open challenges

Characterizing σ-additivity; representing more measures with the same numerosity (e.g. Hausdorff measures); applications to probability (e.g. to the modelization of conditional probability in a way that avoids the Borel-Kolmogorov Paradox).

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Open challenges

Characterizing σ-additivity; representing more measures with the same numerosity (e.g. Hausdorff measures); applications to probability (e.g. to the modelization of conditional probability in a way that avoids the Borel-Kolmogorov Paradox). Constructive numerosities?

Elementary numerosity and measures Emanuele Bottazzi Preliminary notions The main result Open challenges

Open challenges

Characterizing σ-additivity; representing more measures with the same numerosity (e.g. Hausdorff measures); applications to probability (e.g. to the modelization of conditional probability in a way that avoids the Borel-Kolmogorov Paradox). Constructive numerosities? Numerosities with omnific integers?