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Representation of maxitive measures: An overview

  • Paul Poncet
Published/Copyright: February 28, 2017
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 1

Abstract

Idempotent integration is an analogue of Lebesgue integration where σ-maxitive measures replace σ-additive measures. In addition to reviewing and unifying several Radon–Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.

MSC 2010: Primary 28B15; Secondary 03E72; 49J52
Keywords: Idempotent integration; Shilkret integral; Sugeno integral; essential supremum; Radon–Nikodym Theorem; maxitive measures; σ-principal measures; localizable measures; countable chain condition; optimal measures; possibility theory

(Communicated by Anatolij Dvurečenskij)

Acknowledgement

I am grateful to Colas Bardavid who carefully read a preliminary version of the manuscript and made very accurate suggestions. I wish to thank Marianne Akian who made useful remarks and provided a counterexample to [127: Exercise II-3.19.1] inserted as Example 2.10, and Jimmie D. Lawson for his advice and comments. I also thank two anonymous referees who pointed out some missing references in the original manuscript.

Appendix A. Some properties of σ-additive measures

The notions of σ-principal or CCC measures were originally introduced for the study of σ-additive measures. Recall that a σ-additive measure m defined on a σ-algebra ℬ is CCC (resp. σ-principal) if the σ-maxitive measure δm is. Also, following Segal [121], m is localizable if, for all σ-ideals ℐ of ℬ, there exists some L ∈ ℬ such that

  1. m(S \ L) = 0, for all S ∈ ℐ;

  2. if there is some B ∈ ℬ such that m(S \ B) = 0 for all S ∈ ℐ, then m(L \ B) = 0.

The next theorem establishes a link between these notions for σ-additive measures. It enlightens the fact that being finite is a very strong condition for a σ-additive measure (while it is of little consequence for a σ-maxitive measure).

Theorem A.1

Let (E, ℬ) is a measurable space and m be a σ-additive measure on ℬ. Consider the following assertions:

  1. m is finite,

  2. m is σ-finite,

  3. m is σ-principal,

  4. m is CCC,

  5. m is localizable.

Then (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5). Moreover, (4) ⇒ (3) under Zorn’s lemma.

Sketch of the Proof

Assume that m is finite, and let us show that m is σ-principal. Let ℐ be a σ-ideal of ℬ. Let a = sup{m(S) : S ∈ ℐ}. We can find some sequence Sn ∈ ℐ such that m(Sn) ↑ a. Defining L := ∪nSn ∈ ℐ, we have m(L) = a. If there exists some S ∈ ℐ such that m(S \ L) > 0, then m(S ∪ L) > a (since m is finite), which contradicts S ∪ L ∈ ℐ. Thus, m(S \ L) = 0, for all S ∈ ℐ, which gives σ-principality of m. The other implications in Theorem A.1 can be proved along the same lines as for σ-maxitive measures.

Appendix B. Residual semigroups

An ordered semigroup is a semigroup (S, ⊙) equipped with a partial order ⩽ compatible with the structure of semigroup, i.e., such that r ⩽ s and r′ ⩽ s′ imply r ⊙ r′ ⩽ s ⊙ s′.

If (S, ⊙) is an ordered semigroup and r, s ∈ S, we say that r is absolutely continuous with respect to s, written r ≪⊙ s, if there exists some t ∈ S such that r ⩽ t ⊙ s. We say that S (or ⊙) is residual if for all r, s ∈ S with r ≪⊙ s, there is an element of S denoted by (r/s)⊙ such that r ⩽ t ⊙ s ⇔ (r/s)⊙ ⩽ t, for all t ∈ S. Note that in this situation we have r ⩽ (r/s)⊙ ⊙ s. A residual semigroup (S, ⊙) is exact if r = (r/s)⊙ ⊙ s for all r, s ∈ S with r ≪⊙ s.

Examples B.1

In R¯+ here is what we have for different choices of semigroup binary operations (recall that ⊕ denotes the maximum and ∧ the minimum):

  • r ≪× s ⇔ (r = s = 0 or s ≠ 0), in which case (r/s)× × s = r. So (R¯+, ×) is an exact residual semigroup.

  • r ≪+ s always holds, and (r/s)+ = 0 ⊕ (r – s). So (R¯+, +) is a non-exact residual semigroup.

  • r ≪⊕ s always holds, and (r/s)⊕ = 0 if r ⩽ s, (r/s)⊕ = r otherwise. So (R¯+, ⊕) is a non-exact residual semigroup.

  • r ≪∧ s ⇔ r ⩽ s, in which case (r/s)∧ = r, so (R¯+, ∧) is an exact residual semigroup.

Proposition B.2

Let (S, ⊙) be an ordered semigroup. If S is residual, then for all nonempty subsets T of S with infimum and all s ∈ S, {t ⊙ s : t ∈ T} has an infimum and

inft∈T(t⊙s)=(infT)⊙s. (9)

Conversely, if every non-empty subset of $S$ has an infimum and Equation (9) is satisfied for all nonempty subsets T of S with infimum and all s ∈ S, then S is residual.

Proof

First assume that S is residual. Let T be a nonempty subset of S with infimum, and let s ∈ S. Then (inf T) ⊙ s is a lower-bound of the set A = {t ⊙ s : t ∈ T}. Now let ℓ be a lower-bound of A. Since T is non-empty we have ℓ ≪⊙ s. Moreover, ℓ ⩽ t ⊙ s for all t ∈ T, so that (ℓ/s)⊙ ⩽ t for all t ∈ T. This shows that (ℓ/s)⊙ ⩽ inf T, i.e., that ℓ ⩽ (inf T) ⊙ s. So (inf T) ⊙ s is the greatest lower bound of A, i.e., its infimum, and we have proved Equation (9).

Conversely, assume that every non-empty subset of S has an infimum and that Equation (9) is satisfied, and let r, s ∈ S such that r ≪⊙ s. Define (r/s)⊙ = inf T, where T is the nonempty set {t ∈ S : r ⩽ t ⊙ s}. Thanks to Equation (9), the equivalence r ⩽ t ⊙ s ⇔ (r/s)⊙ ⩽ t, for all t ∈ S, is now obvious. So S is residual.

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Received: 2014-06-11
Accepted: 2015-05-14
Published Online: 2017-02-28
Published in Print: 2017-03-01

© 2017 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. State hoops
  2. On derivations of partially ordered sets
  3. Interior and closure operators on commutative basic algebras
  4. When is the cayley graph of a semigroup isomorphic to the cayley graph of a group
  5. Sequences of cantor type and their expressibility
  6. δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials
  7. Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3
  8. Closed hereditary coreflective subcategories in epireflective subcategories of Top
  9. Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments
  10. Difference of two strong Światkowski lower semicontinuous functions
  11. Fejér-type inequalities (II)
  12. Representation of maxitive measures: An overview
  13. On a conjecture of Y. H. Cao and X. B. Zhang
  14. On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces
  15. Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups
  16. Homological properties of banach modules over abstract segal algebras
  17. Variable Hajłasz-Sobolev spaces on compact metric spaces
  18. Commuting pairs of self-adjoint elements in C*-algebras
  19. Additivity of maps preserving products AP ± PA* on C*-algebras
  20. A note on derived connections from semi-symmetric metric connections
  21. Lindelöf P-spaces need not be Sokolov
  22. Strong convergence properties for arrays of rowwise negatively orthant dependent random variables
  23. Least absolute deviations problem for the Michaelis-Menten function
  24. Congruence pairs of principal p-algebras
https://doi.org/10.1515/ms-2016-0253
Keywords for this article
Idempotent integration; Shilkret integral; Sugeno integral; essential supremum; Radon–Nikodym Theorem; maxitive measures; σ-principal measures; localizable measures; countable chain condition; optimal measures; possibility theory

Articles in the same Issue

  1. State hoops
  2. On derivations of partially ordered sets
  3. Interior and closure operators on commutative basic algebras
  4. When is the cayley graph of a semigroup isomorphic to the cayley graph of a group
  5. Sequences of cantor type and their expressibility
  6. δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials
  7. Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3
  8. Closed hereditary coreflective subcategories in epireflective subcategories of Top
  9. Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments
  10. Difference of two strong Światkowski lower semicontinuous functions
  11. Fejér-type inequalities (II)
  12. Representation of maxitive measures: An overview
  13. On a conjecture of Y. H. Cao and X. B. Zhang
  14. On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces
  15. Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups
  16. Homological properties of banach modules over abstract segal algebras
  17. Variable Hajłasz-Sobolev spaces on compact metric spaces
  18. Commuting pairs of self-adjoint elements in C*-algebras
  19. Additivity of maps preserving products AP ± PA* on C*-algebras
  20. A note on derived connections from semi-symmetric metric connections
  21. Lindelöf P-spaces need not be Sokolov
  22. Strong convergence properties for arrays of rowwise negatively orthant dependent random variables
  23. Least absolute deviations problem for the Michaelis-Menten function
  24. Congruence pairs of principal p-algebras
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