, University of Trento January 24-25, 2013, Pisa
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. The measure (Ω, A, µ) is called non-atomic when all finite sets in A have measure zero.
Finitely additive measures: a motivating example For all half-open intervals [x, x + a) R, we can define the length of the interval by posing: L([x, x + a)) = a This function, defined on the family I of half-open intervals, can be extended on the ring B generated by I (i.e. the elements of B are finite unions of half-open intervals). (R, B, L) is then a pre-measure with the additional property that all non-empty sets in B have positive measure. Its extension by the usual Caratheodory construction yelds the Lebesgue measure over R.
Non-Archimedean 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. Definition A superreal field is an ordered field F that properly extends R. Proposition Any superreal field F is non-archimedean, so it contains infinite and (nonzero) infinitesimal numbers.
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) + ɛ where sh(ξ) R and ɛ = ξ sh(ξ) is infinitesimal. Moreover, sh(ξ + ζ) = sh(ξ) + sh(ζ) and sh(ξ ζ) = sh(ξ) sh(ζ). We also define sh(ξ) = + whenever ξ is positive infinite, and sh(ξ) = whenever ξ is negative infinite.
numerosities Definition An elementary 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 non-archimedean 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 Proposition Let n : P(Ω) [0, + ) F be an elementary, and for every β > 0 in F define the function n β : P(Ω) [0, + ] R by posing ( ) n(a) 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.
result (1) Theorem Let (Ω, A, µ) be a non-atomic finitely additive measure. Then there exist a non-archimedean field F R ; an elementary n : P(Ω) [0, + ) F ; such that for every positive number of the form β = n(a ) µ(a ) one has µ(a) = n β (A) for all A A. Moreover, we get the following invariance result: if B A is a subring whose non-empty sets have all positive measure, then we can also assume that n(b) = n(b ) for all B, B B such that µ(b) = µ(b ).
result (2) Theorem Let A be a ring of subsets of Ω and let µ : A [0, + ] R be a non-atomic pre-measure. Then, along with the associated outer measure µ, there exists an inner finitely additive measure such that: µ : P(Ω) [0, + ] R 1 There exists an elementary n : P(Ω) F such that µ = n β for every positive β of the form β = n(a ) µ(a ). 2 µ(c) = µ(c) for all C C µ, the Caratheodory σ-algebra associated to µ. 3 µ(x ) µ(x ) for all X Ω.
Application to Lebesgue measure Corollary Let (R, L, µ L ) be the Lebesgue measure over R. Then there exists an elementary n : P(R) F such that: 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. ( ) sh n(x ) n([0,1)) = µ L (X ) for all X L. ) sh µ L (X ) for all X R. ( n(x ) n([0,1))
An application to probability Corollary Let (Ω, A, µ) be the Kolmogorov probability for infinite coin tosses. There exists an elementary n : P(Ω) F such that the function P(A) = n(a)/n(ω) satisfies the conditions: 1 sh(p(a)) = µ(a) for all A A 2 P agrees with µ over all cylindrical sets: ( ) ( ) P = µ C (i 1,...,i n) (t 1,...,t n) C (i 1,...,i n) (t 1,...,t n) = 2 n 3 if F Ω is finite, then for all A Ω, P(A F ) = A F F
Some ideas for further research From measures to integrals; representing more measures with the same (e.g. Hausdorff measures); applications to probability (e.g. to the modelization of conditional probability in a way that avoids the Borel-Kolmogorov Paradox).
Selected references V. Benci, E., M. Di Nasso,, http://arxiv.org/abs/1212.6201 V. Benci, M. Di Nasso (2003), Numerosities of labelled sets: a new way of counting, Advances in Mathematics, vol. 173, pp. 50 67. C. W. Henson (1972), On the nonstandard representation of measures, Transactions of the American Mathematical Society, vol. 172, pp. 437 446. F. Wattenberg (1977), Nonstandard measure theory - Hausdorff measure, Proceedings of the American Mathematical Society, vol. 65, pp. 326 331.