**6. Main results**

For each *B* in **B**, denote by *s* the Hausdorff dimension of *B* then the Hausdorff *s*-dimensional outer measure is called the Hausdorff outer measure *associated* with the coherent upper prevision *P*(·|*B*). Let *B* ∈ **B** be meausurable with respect to the Hausdorf outer measure associated with *P*(·|*B*).

The Choquet integral representation of a coherent upper conditional prevision with respect to its associated Hausdorff outer measure has bees investigated in [7]. In [9] necessary and sufficient conditions are given such that a coherent upper conditional prevision is uniquely represented as the Choquet integral with respect to its associated Hausdorff outer measure.

In [9, Theorem 4] it is proven that, if the conditioning event has positive and finite Hausdorff outer measure in its dimension *s* and **K**(*B*) is a linear lattice of bounded random variables defined on *B*, necessary conditions for the functional *P*(*X*|*B*) to be represented as Choquet integral with respect to the upper conditional probability *μ*∗ *<sup>B</sup>*, i.e. *<sup>P</sup>*(*X*|*B*) = <sup>1</sup> *hs*(*B*) *Xdhs*, are

**Example 2.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be the linear space of all bounded Borel-measurable random variables on *B* and let *S* be the Borel *σ*-field of subsets of *B*. Denote by *s* the Hausdorff dimension of the conditioning event *B* and by *h<sup>s</sup>* the Hausdorff *s*-dimensional outer measure. If 0 < *hs*(*B*) <

on *S* since each Hausdorff *s*-dimensional (outer) measure is *σ*-additive on the Borel *σ*-field . Moreover let *P*(·|*B*) be a coherent linear conditional prevision, which is continuous from below. Then *P*(·|*B*) can be uniquely represented as the Choquet integral with respect to the

The previous example can be obtained as a consequence of the Daniell-Stone Representation

In this chapter a model of coherent upper conditional precision is introduced. It is defined by the Choquet integral with respect to the *s*-dimensional Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension *s*. Otherwise if the conditioning event has Hausdorff outer measure in its Hausdorff dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension the given upper conditional prevision, defined on a linear lattice of bounded random variables which contains all constants, is uniquely represented as the Choquet integral with respect Hausdorff outer measure if and only if it is a functional which is monotone, submodular, comonotonically additive and continuous

Coherent upper conditional prevision based on the Hausdorff *s*-dimensional measure permits to analyze complex systems where information represented by sets with Hausdorff dimension less than *s*, have no influence on the situation; information represented by sets with the same

Coherent upper previsions defined by Hausdorff outer measures can also be applied in decision theory, to asses preferences between random variables defined on fractal sets and

[1] P. Artzner, F. Delbaen, J. Elber, D. Heath.(1999) Coherent measures of risk. Mathematical

Hausdorff dimension of the conditioning event can influence the system.

*Department of Engineering and Geology, University G.d'Annunzio, Chieti-Pescara, Italy*

*<sup>P</sup>*(*X*|*B*) = *Xdμ<sup>B</sup>* <sup>=</sup> <sup>1</sup>

*hs*(*B*) , for every *A* ∈ *S*; *μB*(*A*) is modular and continuous from below

Coherent Upper Conditional Previsions De ned by

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

61

*Xdhs*.

*hs*(*B*)

<sup>+</sup><sup>∞</sup> define *<sup>μ</sup>B*(*A*) = *<sup>h</sup><sup>s</sup>*(*AB*)

Theorem [5, p. 18].

**7. Conclusions**

from below.

**Author details**

**8. References**

Serena Doria

to defined coherent risk measures.

Finance, 9, 203-228.

coherent upper conditional probability *μB*, that is

that *P*(*X*|*B*) is monotone, comonotonically additive, submodular and continuous from below.

**Theorem 4.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every <sup>B</sup>* <sup>∈</sup> *<sup>B</sup> denote by s the Hausdorff dimension of the conditioning event B and by h<sup>s</sup> the Hausdorff s-dimensional outer measure. Let K*(*B*) *be a linear lattice of bounded random variables defined on B. If the conditioning event B has positive and finite Hausdorff s-dimensional outer measure then the coherent upper conditional prevision P*(·|*B*) *defined on K*(*B*) *as in Theorem 2 is:*


Moreover if the conditioning event *B* has positive and finite Hausdorff *s*-dimensional outer measure, from the properties of the Choquet integral ([5, Proposition 5.1]) the coherent upper conditional prevision *P*(·|*B*) is


So the functional *P*(·|*B*) can be used to defined a *coherent risk measure* [1]. since it is monotone, subadditive, translation invariant and positively homogeneous.

In [9, Theorem 6] sufficient conditions are given for a coherent upper conditional prevision to be uniquely represented as Choquet intergral with respect to its associated Hausdorff outer measure.

**Theorem 5.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every B* <sup>∈</sup> *<sup>B</sup> denote by s the Hausdorff dimension of the conditioning event B and by h<sup>s</sup> the Hausdorff s-dimensional outer measure. Let K*(*B*) *be a linear lattice of bounded random variables on B containing all constants. If B has positive and finite Hausdorff outer measure in its dimension and the coherent upper conditional prevision P*(·|*B*) *on K*(*B*) *is monotone, comonotonically additive, submodular and continuous from below then P*(·|*B*) *is representable as Choquet integral with respect to a monotone, submodular set function which is continuous from below. Furthermore all monotone set functions on* ℘(*B*) *with these properties agree on the set system of weak upper level sets M* = {{*X* ≥ *x*} |*X* ∈ *K*(*B*), *x* ∈ �} *with the upper conditional probability μ*<sup>∗</sup> *<sup>B</sup>*(*A*) = *<sup>h</sup><sup>s</sup>*(*AB*) *hs*(*B*) *for A* ∈ ℘(*B*)*. Let β be a monotone set function on* ℘(*B*)*, which is submodular, continuous from below and such that represents P*(·|*B*) *as Choquet integral. Then the following equalities hold*

$$\overline{P}(X|B) = \int\_B X d\beta = \int\_B X d\mu\_B^\* = \frac{1}{h^s(B)} \int\_B X d\hbar^s.$$

An example is given in the particular case where **K**(*B*) is the linear space of all bounded Borel-measurable random variables on *B* and the restriction of the Hausdorff *s*-dimensional outer measure to the Borel *σ*-field of subsets of *B* is considered.

**Example 2.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be the linear space of all bounded Borel-measurable random variables on *B* and let *S* be the Borel *σ*-field of subsets of *B*. Denote by *s* the Hausdorff dimension of the conditioning event *B* and by *h<sup>s</sup>* the Hausdorff *s*-dimensional outer measure. If 0 < *hs*(*B*) < <sup>+</sup><sup>∞</sup> define *<sup>μ</sup>B*(*A*) = *<sup>h</sup><sup>s</sup>*(*AB*) *hs*(*B*) , for every *A* ∈ *S*; *μB*(*A*) is modular and continuous from below on *S* since each Hausdorff *s*-dimensional (outer) measure is *σ*-additive on the Borel *σ*-field . Moreover let *P*(·|*B*) be a coherent linear conditional prevision, which is continuous from below. Then *P*(·|*B*) can be uniquely represented as the Choquet integral with respect to the coherent upper conditional probability *μB*, that is

$$P(X|B) = \int X d\mu\_B = \frac{1}{h^s(B)} \int X dh^s.$$

The previous example can be obtained as a consequence of the Daniell-Stone Representation Theorem [5, p. 18].
