Elementary numerosities and measures notions The main result Open - PowerPoint PPT Presentation

Elementary numerosity and measures Emanuele Bottazzi Preliminary Elementary numerosities and measures notions The main result Open Emanuele Bottazzi, University of Trento challenges Naples - Konstanz Model Theory Days 2013

Finitely additive measures Elementary numerosity and measures Definition Emanuele Bottazzi A finitely additive measure is a triple (Ω , A , µ ) where: Preliminary The space Ω is a non-empty set; notions The main A is a ring of sets over Ω, i.e. a non-empty family of result subsets of Ω satisfying the conditions: Open challenges 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.

Finitely additive measures Elementary numerosity and measures Definition Emanuele Bottazzi A finitely additive measure is a triple (Ω , A , µ ) where: Preliminary The space Ω is a non-empty set; notions The main A is a ring of sets over Ω, i.e. a non-empty family of result subsets of Ω satisfying the conditions: Open challenges 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.

Superreal Fields Elementary numerosity and measures Definition Emanuele Bottazzi 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 Preliminary notions if | ξ | > n for all n ∈ N . The main result Open challenges

Superreal Fields Elementary numerosity and measures Definition Emanuele Bottazzi 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 Preliminary notions if | ξ | > n for all n ∈ N . The main result Proposition Open challenges There exist ordered fields that properly extend R . Such fields are non-archimedean, i.e. they contain infinite and (nonzero) infinitesimal numbers.

Superreal Fields Elementary numerosity and measures Definition Emanuele Bottazzi 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 Preliminary notions if | ξ | > n for all n ∈ N . The main result Proposition Open challenges 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 .

The Standard Part Elementary numerosity and measures Emanuele Bottazzi Proposition Preliminary Let F be a superreal field. Every finite number ξ ∈ F can be notions represented in a unique way by The main result Open ξ = sh ( ξ ) + ǫ challenges where sh ( ξ ) ∈ R and ǫ = ξ − sh ( ξ ) is infinitesimal.

The Standard Part Elementary numerosity and measures Emanuele Bottazzi Proposition Preliminary Let F be a superreal field. Every finite number ξ ∈ F can be notions represented in a unique way by The main result Open ξ = sh ( ξ ) + ǫ challenges 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 numerosities Elementary numerosity and measures Emanuele Definition Bottazzi An elementary numerosity on a set Ω is a function Preliminary notions The main n : P (Ω) → [0 , + ∞ ) F result Open challenges 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.

Numerosity measures Elementary numerosity and measures Emanuele Bottazzi Proposition Preliminary Let n : P (Ω) → [0 , + ∞ ) F be an elementary numerosity, and for notions every β > 0 in F define the function n β : P (Ω) → [0 , + ∞ ] R by The main result posing Open � n ( A ) � challenges n β ( A ) = sh . β Then n β is a finitely additive measure defined for all subsets of Ω . Moreover, n β is non-atomic if and only if β is an infinite number.

The main result Elementary numerosity Theorem and measures Let (Ω , A , µ ) be a non-atomic finitely additive measure. Then Emanuele Bottazzi there exist Preliminary a non-archimedean field F ⊇ R ; notions The main an elementary numerosity n : P (Ω) → [0 , + ∞ ) F ; result Open such that for every positive number of the form β = n ( X ) µ ( X ) one challenges has µ ( A ) = n β ( A ) for all A ∈ A .

The main result Elementary numerosity Theorem and measures Let (Ω , A , µ ) be a non-atomic finitely additive measure. Then Emanuele Bottazzi there exist Preliminary a non-archimedean field F ⊇ R ; notions The main an elementary numerosity n : P (Ω) → [0 , + ∞ ) F ; result Open such that for every positive number of the form β = n ( X ) µ ( X ) one challenges has µ ( A ) = n β ( 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 ′ ) .

An application: Lebesgue measure Elementary numerosity and measures Emanuele Corollary Bottazzi Let ( R , L , µ L ) be the Lebesgue measure over R . There exists Preliminary notions an elementary numerosity n : P ( R ) → F such that: The main result Open challenges

An application: Lebesgue measure Elementary numerosity and measures Emanuele Corollary Bottazzi Let ( R , L , µ L ) be the Lebesgue measure over R . There exists Preliminary notions an elementary numerosity n : P ( R ) → F such that: The main result � � n ( A ) sh = µ L ( A ) for all A ∈ L . n ([0 , 1)) Open challenges

An application: Lebesgue measure Elementary numerosity and measures Emanuele Corollary Bottazzi Let ( R , L , µ L ) be the Lebesgue measure over R . There exists Preliminary notions an elementary numerosity n : P ( R ) → F such that: The main result � � n ( A ) sh = µ L ( A ) for all A ∈ L . n ([0 , 1)) Open challenges � � n ( A ) sh ≤ µ L ( A ) for all A ⊆ R . n ([0 , 1))

An application: Lebesgue measure Elementary numerosity and measures Emanuele Corollary Bottazzi Let ( R , L , µ L ) be the Lebesgue measure over R . There exists Preliminary notions an elementary numerosity n : P ( R ) → F such that: The main result � � n ( A ) sh = µ L ( A ) for all A ∈ L . n ([0 , 1)) Open challenges � � n ( A ) sh ≤ µ L ( A ) for all A ⊆ R . n ([0 , 1)) n ([ x , x + a )) = n ([ y , y + a )) for all x , y ∈ R and for all a > 0 .

An application: Lebesgue measure Elementary numerosity and measures Emanuele Corollary Bottazzi Let ( R , L , µ L ) be the Lebesgue measure over R . There exists Preliminary notions an elementary numerosity n : P ( R ) → F such that: The main result � � n ( A ) sh = µ L ( A ) for all A ∈ L . n ([0 , 1)) Open challenges � � n ( A ) sh ≤ µ L ( A ) for all A ⊆ R . n ([0 , 1)) 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 .

An application: the fair coin measure Elementary Corollary numerosity and measures Let (2 N , A , µ ) be the fair coin measure. There exists an Emanuele Bottazzi elementary numerosity n : P (2 N ) → F such that the function Preliminary P ( A ) = n ( A ) / n (2 N ) satisfies the conditions: notions The main result Open challenges

An application: the fair coin measure Elementary Corollary numerosity and measures Let (2 N , A , µ ) be the fair coin measure. There exists an Emanuele Bottazzi elementary numerosity n : P (2 N ) → F such that the function Preliminary P ( A ) = n ( A ) / n (2 N ) satisfies the conditions: notions The main 1 sh ( P ( A )) = µ ( A ) for all A ∈ A result Open challenges

An application: the fair coin measure Elementary Corollary numerosity and measures Let (2 N , A , µ ) be the fair coin measure. There exists an Emanuele Bottazzi elementary numerosity n : P (2 N ) → F such that the function Preliminary P ( A ) = n ( A ) / n (2 N ) satisfies the conditions: notions The main 1 sh ( P ( A )) = µ ( A ) for all A ∈ A result 2 P agrees with µ over all cylindrical sets: Open challenges � � � � C ( i 1 ,..., i n ) C ( i 1 ,..., i n ) = 2 − n P = µ ( t 1 ,..., t n ) ( t 1 ,..., t n )

An application: the fair coin measure Elementary Corollary numerosity and measures Let (2 N , A , µ ) be the fair coin measure. There exists an Emanuele Bottazzi elementary numerosity n : P (2 N ) → F such that the function Preliminary P ( A ) = n ( A ) / n (2 N ) satisfies the conditions: notions The main 1 sh ( P ( A )) = µ ( A ) for all A ∈ A result 2 P agrees with µ over all cylindrical sets: Open challenges � � � � C ( i 1 ,..., i n ) C ( i 1 ,..., i n ) = 2 − n P = µ ( t 1 ,..., t n ) ( t 1 ,..., t n ) 3 if F ⊂ 2 N is finite, then for all A ⊆ 2 N , P ( A | F ) = | A ∩ F | | F |