Elementary numerosity and measures

In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite cardinality. We investigate the relationships between this notion and the notion of measure. The main result is that every non-atomic finitely additive measure is obtained from a suitable elementary numerosity by simply taking its ratio to a unit. In the last section we give applications to this result.


Introduction
In mathematics there are essentially two main ways to estimate the size of a set, depending on whether one is working in a discrete or in a continuous setting.
In the continuous case, one uses the notion of (finitely additive) measure, namely a function m taking real values and which satisfies the following properties: (1) m(∅) = 0 The goal of this paper is to investigate the relationships between these two notions. Of course, this problem is interesting when the sets are infinite. Remark that the theory of infinite cardinality is not adequate to this end; for example, all sets of reals with positive Lebesgue measure have the same cardinality. On the contrary, the notion of numerosity, first introduced in [1,2], gives a coherent way of extending finite cardinalities and their main properties to infinite sets.
In this paper we introduce the related concept of elementary numerosity as a special function defined on all subsets of a given set X that takes values into a suitable ordered field F and satisfies the four properties of finite cardinalities itemized above. Remark that if X is infinite, then the range of such a function n necessarily contains infinite numbers, and hence the field F must be non-Archimedean. Notice that also Cantorian cardinality satisfies properties (n.1), (n.2), (n.3), (n.4); the fundamental difference is that "numerosities" are required to be elements of a field.
By taking ratios to a fixed "measure unit" β > 0, one has a canonical way of getting a real-valued finitely additive measure. This construction turns out to be really general. In fact, the main result of this paper shows that every finitely additive non-atomic measure can be obtained in this way. Namely, we shall prove the following: 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 ; • a positive number β ∈ F such that The last part of the paper is devoted to selected applications: the first one is about Lebesgue measure, and the second one is about non-Archimedean probability. In particular, following ideas from [4], we consider a model for infinite sequences of coin tosses which is coherent with Laplace original view. Indeed, probability of an event is defined as the numerosity of positive outcomes divided by the numerosity of all possible outcomes; moreover, the probability of cylindrical sets exactly coincides with the usual Kolmogorov probability.

Terminology and preliminary notions
We fix here our terminology, and recall a few basic facts from measure theory that will be used in the sequel.
Let us first agree on notation. We write A ⊆ B to mean that A is a subset of B, and we write A ⊂ B to mean that A is a proper subset of B. The complement of a set A is denoted by A c , and its powerset is denoted by P(A). We write A 1 ⊔ . . . ⊔ A n to denote a disjoint union. By N we denote the set of positive integers, and by N 0 = N ∪ {0} the set of non-negative integers. For an ordered field F, we denote by [0, ∞) F = {x ∈ F | x ≥ 0} the set of its non-negative elements. We shall write [0, +∞] R to denote the set of non-negative real numbers plus the symbol +∞, and we agree that x + ∞ = +∞ + x = +∞ + ∞ = +∞ for all x ∈ R. Definition 1.1. 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 which is closed under finite unions and intersections, and under relative complements, i.e. A, whenever A, B ∈ A are disjoint. 3 We also assume that µ(∅) = 0.
The measure (Ω, A, µ) is called non-atomic when all finite sets in A have measure zero. We say that (Ω, A, µ) is a probability when µ : A → [0, 1] R takes values in the unit interval.
For simplicity, in the following we shall often identify the triple (Ω, A, µ) with the function µ.
Remark that a finitely additive measure µ is necessarily monotone, i.e.
2 Actually, the closure under intersections follow from the other two properties, since . 3 Such functions µ are sometimes called contents in the literature.
A measure is a pre-measure which is defined on a σ-algebra, i.e. on a ring of sets which is closed under countable unions and intersections. It is also assumed that M (∅) = 0.
Definition 1.4. Given an outer measure M on Ω, the following family is called the Caratheodory σ-algebra associated to M : A well known theorem of Caratheodory states that the above family is actually a σ-algebra, and that the restriction of M to C M is a complete measure, i.e. a measure where M (X) = 0 implies Y ∈ C M for all Y ⊆ X. This result is usually combined with the property that every pre-measure µ over a ring A of subsets of Ω is canonically extended to the outer measure µ : P(Ω) → [0, ∞] R defined by putting: Indeed, a fundamental result in measure theory is that the above function µ is actually an outer measure that extends µ, and that the associated Caratheodory σ-algebra C µ includes A. Moreover, such an outer measure µ is regular, i.e. for all X ∈ P(Ω) there exists C ∈ C µ such that X ⊆ C and µ(X) = µ(C). (See e.g. [8] Prop. 20.9.) In the proof of our main theorem, we shall use an ultrapower R I /U of the real numbers modulo a suitable ultrafilter. Recall that an ultrafilter U on a set I is a maximal family of subsets of I which has the finite intersection property (FIP): A 1 ∩ . . . ∩ A n = ∅ for any choice of finitely many A i ∈ U . Equivalently, U is a family of non-empty subsets of I that is closed under supersets, finite intersections, and satisfies the property A / ∈ U ⇒ I \ A ∈ U . Remark that an ultrafilter U can also be characterized as a family of sets that have measure 1 with respect to a suitable finitely additive {0, 1}-valued measure µ : P(I) → {0, 1}. By applying Zorn's lemma, it is shown that every family F ⊆ P(I) with the FIP can be extended to an ultrafilter on I.
The ultrapower F = R I /U of the real numbers modulo the ultrafilter U is the ordered field where: • Elements of F are the real I-sequences σ U defined U -almost everywhere, i.e. σ U = τ U when {i ∈ I | σ(i) = τ (i)} ∈ U ; • The order relation and the sum and product operations are defined For detailed information about ultrafilters and the general construction of ultrapower, the reader is referred to e.g. [6].

Elementary numerosity
Inspired by the idea of numerosity, we now aim at refining the notion of finitely additive measure in such a way that also single points count. To this end, one needs to consider superreal fields F ⊇ R, i.e. ordered fields which extend the real line.
Remark that if the field F ⊃ R is a proper extension, then F is necessarily non-Archimedean, i.e. it contains infinitesimal numbers ǫ = 0 such that −1/n < ǫ < 1/n for all n ∈ N. We say that two elements ξ, ζ ∈ F are infinitely close, and write ξ ≈ ζ, when their difference ξ − ζ is infinitesimal. A number ξ ∈ F is called finite when −n < ξ < n for some n ∈ N, and it is called infinite otherwise. Clearly, a number ξ is infinite if and only if its reciprocal 1/ξ is infinitesimal. Since F ⊇ R, by the completeness property of the real line it is easily verified that every finite ξ ∈ F is infinitely close to a unique real number r (just take r = inf{x ∈ R | x > ξ}). Such a number r is called the shadow (or standard part) of ξ, and notation r = sh(ξ) is used. Notice that sh(ξ + ζ) = sh(ξ) + sh(ζ) and sh(ξ · ζ) = sh(ξ) · sh(ζ) for all finite ξ, ζ. By abusing notation, we shall write sh(ξ) = +∞ when ξ is infinite and positive, and sh(ξ) = −∞ when ξ is infinite and negative. defined for all subsets of Ω, taking values into the non-negative part of a superreal field F, and such that the following two conditions are satisfied: As straight consequences of the definition, we obtain that elementary numerosities can be seen as generalizations of finite cardinalities. Proof. Notice that n(∅) = n(∅ ∪ ∅) = n(∅) + n(∅), and x = 0 is the only number In consequence, µ(A) > 0 for all non-empty sets A. Finally, the last property directly follows by additivity and the fact that every singleton has measure 1.
Remark that if one takes F = R then elementary numerosities n exist on a set Ω if and only if Ω is finite and n is the finite cardinality. However, we shall see that our assumption which allows n to take non-Archimedean values, will make it possible to extend the "counting measure" as given by finite cardinality, to arbitrary infinite sets.
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.
Proof. For all disjoint A, B ⊆ Ω, one has: Notice that the measure n β is non-atomic if and only if n β ({x}) = sh(1/β) = 0, and this holds if and only if β is infinite.
The above class of measures turns out to be really general. In the next section we shall show that every finitely additive non-atomic measure is in fact a restriction of a suitable n β .

The main result
Moreover, if B ⊆ A is a subring whose non-empty sets have all positive measure, then we can also assume that Proof. Denote by A f (by B f ) the set of all elements of A (of B, respectively) which have finite measure. Let Λ = Fin(Ω) be the family of all finite subsets of Ω, and define the following sets.
• For all x ∈ Ω, let Then consider the following family of subsets of Λ: We want to show that all finite intersections of elements of G are nonempty. To this end, we shall use the following combinatorial result, whose proof is put off to the Appendix. Lemma 3.2. Let (Ω, A, µ) be a non-atomic finitely additive measure, and let B ⊆ A be a subring of subsets of Ω whose non-empty sets have all positive finite measure. Given m ∈ N, given finitely many points x 1 , . . . , x k ∈ Ω, and given finitely many non-empty sets A 1 , . . . , A n ∈ A having finite measure, there exists a finite subset λ ⊂ Ω that satisfies the following properties: Now let finitely many elements of G be given, say . Pick m = max{n 1 , . . . , n u } and apply the above Lemma to get the existence a finite set λ ∈ Λ such that (1) x 1 , . . . , x k ∈ λ ; (2) For all

VIERI BENCI, EMANUELE BOTTAZZI, AND MAURO DI NASSO
Then it is readily verified that such a λ belongs to all considered sets of G. In consequence of this finite intersection property, the family G ⊂ P(Λ) can be extended to an ultrafilter U on Λ. Now define the following: • F = R Λ /U is the ordered field obtained as the ultrapower of R modulo the ultrafilter U . We identify each real number r with the corresponding constant sequence c r U defined U -almost everywhere, so that R ⊆ F. Let us now verify that all the desired properties are satisfied. Given A, A ′ ∈ A f with µ(A ′ ) = 0, for every n ∈ N we have that As this holds for every n, we conclude that As a straight consequence, we obtain the following (3) µ(X) ≤ µ(X) for all X ⊆ Ω.
Proof. By Caratheodory extension theorem, the restriction of µ to C µ is a measure that agrees with µ on A. By applying the previous theorem to µ |Cµ , we obtain the existence of an elementary numerosity n : P(Ω) → [0, +∞) F such that for every positive number of the form β = n(A * ) µ(A * ) one has n β (C) = µ(C) for all C ∈ C µ . We claim that µ = n β : P(Ω) → [0, +∞] R is the desired "inner" finitely additive measure.
Properties (1) and (2) are trivially satisfied by our definition of µ, so we are left to show (3). For every X ⊆ Ω, by definition of outer measure we have that for every ǫ > 0 there exists a countable union A = ∞ n=1 A n of sets A n ∈ A such that A ⊇ X and ∞ n=1 µ(A n ) ≤ µ(X) + ǫ. Notice that A belongs to the σ-algebra generated by A, and hence A ∈ C µ . In consequence, µ(A) = n β (A) = µ(A). Finally, by monotonicity of the finitely additive measure µ, and by σ-subadditivity of the outer measure µ, we obtain: As ǫ > 0 is arbitrary, the desired inequality µ(X) ≤ µ(X) follows.
It seems of some interest to investigate the properties of the estension of the Caratheodory algebra given by family of all sets for which the outer measure coincides with the above "inner measure": Clearly, the properties of C(n β ) may depend on the choice of the elementary numerosity n β . Theorem 3.3 ensures that the inclusion C µ ⊆ C(n β ) always holds. Moreover, this inclusion is an equality if and only if all X ∈ C µ satisfy the inequality µ(X) < µ(X). It turns out that this property is equivalent to a number of other statements. (1) X ∈ C µ ⇒ µ(X) < µ(X) and µ(X c ) < µ(X c ).

Applications
In this last section, we present two consequences of Theorems 3.1 and 3.3 that may have some relevance in applications.  (1) n([x, x + a)) = n([y, y + a)) for all x, y ∈ R and for all a > 0.

Proof. Notice that the family of half-open intervals
generates a subring B ⊂ L whose non-empty sets have all positive measure. Then by combining Theorems 3.1 and 3.3 we obtain the existence of an elementary numerosity n : P(R) → F such that for every positive number of the form β = n(A * ) µ(A * ) one has: (i) n(X) = n(Y ) whenever X, Y ∈ B are such that µ L (X) = µ L (Y ) ; (ii) n β (X) = µ L (X) for all X ∈ L ; (iii) n β (X) ≤ µ L (X) for all X ⊆ R.
Since [x, x + a) ∈ B for all x ∈ R and for all a > 0, property (1) directly follows from (i). In order to prove (2), it is enough to show that n([0, a)) = a · n([0, 1) for all positive a ∈ Q. Given p, q ∈ N, by (1) and additivity we have that In particular, for p = q we get that n([0, 1)) = q · n([0, 1/q)), and hence property (2) follows: Finally, if we take as then (ii) and (iii) correspond to properties (3) and (4), respectively. necessarily has translates t + X with a different n β -measure: n β (t + X) = n β (X). In fact, recall that Lebesgue measure µ L is characterized as the unique translation-invariant measure on the Borel subsets of R such that µ L ([0, 1)) = 1. 4

Elementary numerosities and probability of infinite coin tosses.
The second application of our results on elementary numerosities is about the existence of a non-Archimedean probability for infinite sequences of coin tosses, which we propose as a sound mathematical model for Laplace's original ideas. Recall the Kolmogorovian framework : • The sample space is the set of sequences which take either H ("head") or T ("tail") as values.
• A cylinder set of codimension n is a set of the form: 5 From the probabilistic point of view, the cylinder set C (i 1 ,...,in) (t 1 ,...,tn) represents the event that for all s = 1, . . . , n, the i s -th coin toss gives t s as outcome. Notice that the family C of all cylinder sets is a ring of sets over Ω.
• The function µ C : C → [0, 1] is defined by setting: It is shown that µ C is a pre-measure of probability on the ring C.
• A is the σ-algebra generated by the ring of cylinder sets C ; • µ : A → [0, 1] is the unique probability measure that extends µ C , as guaranteed by Caratheodory extension theorem.
The triple (Ω, A, µ) is named the Kolmogorovian probability for infinite sequences of coin tosses.
In [4], it is proved the existence of an elementary numerosity n : P(Ω) → F which is coherent with the pre-measure µ C . Namely, by considering the ratio P (E) = n(E)/n(Ω) between the numerosity of the given event E and the numerosity of the whole space Ω, then one obtains a non-Archimedean finitely additive probability P : P(Ω) → [0, 1] F that satisfies the following properties: (1) If F ⊂ Ω is finite, then for all E ⊆ Ω, the conditional probability (2) P agrees with µ C over all cylindrical sets: We are now able to refine this result by showing that, up to infinitesimals, we can take P to agree with µ on the whole σ-algebra A. (3) sh(P (E)) = µ(E) for all E ∈ A. 5 We agree that i1 < . . . < in.
Proof. Recall that the family C ⊂ A of cylinder sets is a ring whose nonempty sets have all positive measure. So, by applying Theorems 3.1 and 3.3, we obtain an elementary numerosity n : P(Ω) → F such that for every positive number of the form β = n(A * ) µ(A * ) one has: (i) n(C) = n(C ′ ) whenever C, C ′ ∈ C are such that µ(C) = µ(C ′ ) ; (ii) n β (E) = µ(E) for all E ∈ A.
Appendix A. Proof of Lemma 3.2 Without loss of generality, we can assume that the given sets A i are arranged in such a way that A 1 , . . . , A l ∈ B and A l+1 , . . . , A n ∈ A \ B for a suitable l. It will be convenient in the sequel that the considered elements in B be pairwise disjoint. To this end, consider the partition that, by the ring properties of B, every piece B s belongs to B.) Finally, let be the partition induced by {B 1 , . . . , B h , A l+1 , . . . , A n }, where µ(C s ) > 0 for s = 1, . . . , p and µ(D t ) = 0 for t = 1, . . . , q. For every s = 1, . . . , h, the set B s include at least one piece C j of positive measure in the above partition. Moreover, since B 1 , . . . , B h are pairwise disjoint, by re-arranging if necessary, we can also assume that C s ⊆ B s for s = 1, . . . , h. Now recall the following Dirichlet's simultaneous approximation theorem (see e.g. [7] §11.12): "Given finitely many real numbers y s > 0, for every ε > 0 there exist arbitrarily large numbers N ∈ N such that every fractional part {N · y s } = N · y s − [N · y s ] < ε". So, if we let . . , p} then we can pick a natural number N such that: Notice that N s > k for all s. In fact, by the above conditions (a) and (b): For s = 1, . . . , h, pick a finite subset λ s ⊂ C s containing exactly (N s −M s )many elements, and such that C s ∩ F ⊆ λ s . Observe that this is in fact possible because For s = h+1, . . . , p, pick a finite subset λ s ⊂ C s containing exactly N s -many elements. Finally, define We claim that λ has the desired properties. Condition (1) is trivially is satisfied because F ⊆ λ by definition. For every i = 1, . . . , n let: With the above definitions, we obtain: where: • ε i = s∈G(i)∪G ′ (i) e s ≤ p s=1 e s < 1 by condition (b) ; If A i ∈ B, i.e. if i ≤ l, recall that A i = s∈S(i) B s for a suitable S(i) ⊆ {1, . . . , h}. Since C s ⊆ B s for all s = 1, . . . , h, it must be G(i) = S(i). So, for i ≤ l we have and hence |λ ∩ A i | = N · µ(A i ) − ε i . In consequence, for every i, j ≤ l such that µ(A i ) = µ(A j ), one has that Now notice that |ε j − ε i | ≤ max{ε i , ε j } < 1, and so the natural numbers |λ ∩ A i | = |λ ∩ A j | necessarily coincide. This completes the proof of (2).
So, we reach the thesis: