**3. Separately coherent upper conditional previsions defined by Hausdorff outer measures**

In this section coherent upper conditional previsions are defined by the Choquet integral with respect to Hausdorff outer measures if the conditioning event *B* has positive and finite Hausdorff outer measure in its dimension. Otherwise if the conditioning event *B* has Hausdorff outer measure in its dimension equal to zero or infinity they are defined by a 0-1 valued finitely, but not countably, additive probability.

## **3.1. Hausdorff outer measures**

Given a non-empty set Ω, let ℘(Ω) be the class of all subsets of Ω. An *outer measure* is a function *μ*<sup>∗</sup> : ℘(Ω) → [0, +∞] such that *μ*∗(�) = 0, *μ*∗(*A*) ≤ *μ*∗(*A*� ) if *A* ⊆ *A*� and *μ*∗( ∞ *<sup>i</sup>*=<sup>1</sup> *Ai*) <sup>≤</sup> <sup>∑</sup><sup>∞</sup> *<sup>i</sup>*=<sup>1</sup> *μ*∗(*Ai*).

Examples of outer set functions or outer measures are the Hausdorff outer measures [11], [15].

Let (Ω, *d*) be a metric space. A topology, called the *metric topology*, can be introduced into any metric space by defining the open sets of the space as the sets *G* with the property:

if *x* is a point of *G*, then for some *r* > 0 all points y with *d*(*x*, *y*) < *r* also belong to *G*.

It is easy to verify that the open sets defined in this way satisfy the standard axioms of the system of open sets belonging to a topology [15, p.26].

The Borel *σ*-field is the *σ*-field generated by all open sets. The Borel sets include the closed sets (as complement of the open sets), the *Fσ*-sets (countable unions of closed sets) and the *Gσ*-sets (countable intersections of open sets), etc.

The diameter of a non empty set *U* of Ω is defined as |*U*| = sup {*d*(*x*, *y*) : *x*, *y* ∈ *U*} and if a subset *<sup>A</sup>* of <sup>Ω</sup> is such that *<sup>A</sup>* ⊂ *<sup>i</sup> Ui* and 0 < |*Ui*| < *δ* for each i, the class {*Ui*} is called a *δ*-cover of *A*.

Let *<sup>s</sup>* be a non-negative number. For *<sup>δ</sup>* >0 we define *hs*,*<sup>δ</sup>* (*A*) = inf <sup>∑</sup>+<sup>∞</sup> <sup>ß</sup>=<sup>1</sup> |*Ui*| *s* , where the infimum is over all *δ*-covers {*Ui*}.

The *Hausdorff s-dimensional outer measure* of *A*, denoted by *hs*(*A*), is defined as

$$h^s(A) = \lim\_{\delta \to 0} h\_{s\delta}(A).$$

This limit exists, but may be infinite, since *hs*,*δ*(*A*) increases as *δ* decreases because less *δ*-covers are available. The *Hausdorff dimension* of a set *A*, *dimH*(*A*), is defined as the unique value, such that

$$\begin{aligned} h^s(A) &= +\infty \quad \text{if} \quad 0 \le \text{s} < \text{dim}\_{\mathcal{H}}(\mathcal{A}),\\ h^s(A) &= 0 \quad \text{if} \quad \text{dim}\_{\mathcal{H}}(\mathcal{A}) < \text{s} < +\infty. \end{aligned}$$

We can observe that if 0 < *hs*(*A*) < +∞ then *dimH*(*A*) = *s*, but the converse is not true.

Hausdorff outer measures are *metric* outer measures:

4 Stochastic Control

A bounded random variable is called *B-measurable* or measurable with respect to the partition **B** [19, p.291] if it is constant on the atoms *B* of the partition. Let *G*(**B**) be the class of all

Separately coherent upper conditional previsions *P*(*X*|*B*) can be extended to a common domain **H** so that the function *P*(·|**B**) can be defined from **H** to *G*(**B**) to summarize the

*P*(·|**B**) is assumed to be separately coherent if all the *P*(·|*B*) are separately coherent. In

It is proven that, every time that the *σ*-field of the conditioning events is properly contained in the *σ*-field of the probability space and it contains all singletons, the Radon-Nikodym derivative cannot be used as a tool to define coherent conditional previsions. This is due to the fact that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the *σ*-field of the conditioning events, contradicts a necessary

Analysis done points out the necessity to introduce a different tool to define coherent

**3. Separately coherent upper conditional previsions defined by Hausdorff**

In this section coherent upper conditional previsions are defined by the Choquet integral with respect to Hausdorff outer measures if the conditioning event *B* has positive and finite Hausdorff outer measure in its dimension. Otherwise if the conditioning event *B* has Hausdorff outer measure in its dimension equal to zero or infinity they are defined by a 0-1

Given a non-empty set Ω, let ℘(Ω) be the class of all subsets of Ω. An *outer measure* is

Examples of outer set functions or outer measures are the Hausdorff outer measures [11], [15]. Let (Ω, *d*) be a metric space. A topology, called the *metric topology*, can be introduced into any

It is easy to verify that the open sets defined in this way satisfy the standard axioms of the

The Borel *σ*-field is the *σ*-field generated by all open sets. The Borel sets include the closed sets (as complement of the open sets), the *Fσ*-sets (countable unions of closed sets) and the *Gσ*-sets

) if *A* ⊆ *A*� and

a function *μ*<sup>∗</sup> : ℘(Ω) → [0, +∞] such that *μ*∗(�) = 0, *μ*∗(*A*) ≤ *μ*∗(*A*�

metric space by defining the open sets of the space as the sets *G* with the property: if *x* is a point of *G*, then for some *r* > 0 all points y with *d*(*x*, *y*) < *r* also belong to *G*.

Theorem 1 [9] the function *P*(*X*|**B**) is compared with the Radon-Nikodym derivative.

Denote by *P*(*X*|**B**) the random variable equal to *P*(*X*|*B*) if *ω* ∈ *B*.

valued finitely, but not countably, additive probability.

system of open sets belonging to a topology [15, p.26].

**B**-measurable random variables.

collection of *P*(*X*|*B*) with *B* ∈ **B**.

condition for the coherence.

**3.1. Hausdorff outer measures**

*<sup>i</sup>*=<sup>1</sup> *μ*∗(*Ai*).

(countable intersections of open sets), etc.

*<sup>i</sup>*=<sup>1</sup> *Ai*) <sup>≤</sup> <sup>∑</sup><sup>∞</sup>

conditional previsions.

**outer measures**

*μ*∗( ∞

$$h^s(E \cup F) = h^s(E) + h^s(F) \text{ whenever } d(E, F) = \inf \left\{ d(\mathbf{x}, y) : \mathbf{x} \in E, y \in F \right\} > 0.$$

A subset *A* of Ω is called *measurable* with respect to the outer measure *h<sup>s</sup>* if it decomposes every subset of Ω additively, that is if

$$h^s(E) = h^s(A \cap E) + h^s(E - A) \text{ for all sets } E \subseteq \Omega.$$

All Borel subsets of Ω are measurable with respect to any metric outer measure [11, Theorem 1.5]. So every Borel subset of Ω is measurable with respect to every Hausdorff outer measure *h<sup>s</sup>* since Hausdorff outer measures are metric.

The restriction of *h<sup>s</sup>* to the *σ*-field of *hs*-measurable sets, containing the *σ*-field of the Borel sets, is called Hausdorff s-dimensional measure. In particular the Hausdorff 0-dimensional measure is the counting measure and the Hausdorff 1-dimensional measure is the Lebesgue measure.

The Hausdorff *s*-dimensional measures are *modular* on the Borel *σ*-field, that is

$$h^s(A \cup B) + h^s(A \cap B) = h^s(A) + h^s(B)$$

for every pair of Borelian sets *A* and *B*; so that [5, Proposition 2.4] the Hausdorff outer measures are *submodular*

$$h^s(A \cup B) + h^s(A \cap B) \le h^s(A) + h^s(B).$$

In [15, p.50] and [11, Theorem 1.6 (a)] it has been proven that if *A* is any subset of Ω there is a *Gσ*-set *G* containing *A* with *hs*(*A*) = *hs*(*G*). In particular *h<sup>s</sup>* is an *outer regular* measure.

Moreover Hausdorff outer measures are *continuous from below* [11, Lemma 1.3], that is for any increasing sequence of sets {*Ai*} we have

$$\lim\_{i \to \infty} h^s(A\_i) = h^s(\lim\_{i \to \infty} A\_i).$$

is decreasing and it is called *decreasing distribution function* of *X* with respect to *μ*. If *μ* is continuous from below then *Gμ*,*X*(*x*) is right continuous. In particular the decreasing distribution function of *X* with respect to the Hausdorff outer measures is right continuous since these outer measures are continuous from below. A function *X* : Ω → � is called upper *μ*-measurable if *Gμ*∗,*X*(*x*) = *Gμ*∗,*X*(*x*). Given an upper *μ*-measurable function *X* :Ω → *R* with decreasing distribution function *Gμ*,*X*(*x*), if *μ*(Ω) < +∞, the *asymmetric Choquet integral* of *X*

(*Gμ*,*X*(*x*) − *μ*(Ω))*dx* +

The integral is in �, can assume the values −∞, +∞ or is undefined when the right-hand side

<sup>0</sup> *<sup>G</sup>μ*,*X*(*x*)*dx* <sup>=</sup> sup *<sup>X</sup>*

**Theorem 1.** *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 space of bounded random variables on B. Moreover, let m be a 0-1 valued finitely additive, but not countably additive, probability on* ℘(*B*) *such that a different m is chosen for each B. Then for each B* ∈ *B the functional P*(*X*|*B*) *defined on K*(*B*) *by*

*<sup>P</sup>*(*X*|*B*) = *<sup>m</sup>*(*XB*) *if hs*(*B*) = 0, <sup>+</sup><sup>∞</sup>

The lower conditional previsions *P*(*A*|*B*) can be define as in the previous theorem if *hs* denotes

Given an upper conditional prevision *P*(*X*|*B*) defined on a linear space the lower conditional prevision *P*(*X*|*B*) is obtained as its conjugate, that is *P*(*X*|*B*) = −*P*(−*X*|*B*). If *B* has positive and finite Hausdorff outer measure in its Hausdorff dimension *s* and we denote by *hs* the

*hs*(*B*)

 *B*

(−*X*)*dh<sup>s</sup>* <sup>=</sup>

<sup>0</sup> *Gμ*,*X*(*x*)*dx*

*<sup>B</sup> Xdh<sup>s</sup> if 0* <sup>&</sup>lt; *<sup>h</sup>s*(*B*) <sup>&</sup>lt; <sup>+</sup><sup>∞</sup>

If *X* ≥ 0 or *X* ≤ 0 the integral always exists. In particular for *X* ≥ 0 we obtain

*Xdμ* = <sup>+</sup><sup>∞</sup>

A new model of coherent upper conditional prevision is defined in [9, Theorem 2].

 ∞ 0

*Gμ*,*X*(*x*)*dx*

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

Coherent Upper Conditional Previsions De ned by

57

inf *<sup>X</sup> Gμ*,*X*(*x*)*dx* + inf *X*.

with respect to *μ* is defined by

is +∞ − ∞.

*Xdμ* = <sup>0</sup>

*and by*

**3.3. The model**

If *X* is bounded and *μ*(Ω) = 1 we have that

*is a coherent upper conditional prevision.*

the Hausdorff s-dimensional inner measure.

Hausdorff *s*-dimensional inner measure we have

inf *<sup>X</sup>*(*Gμ*,*X*(*x*) <sup>−</sup> <sup>1</sup>)*dx* <sup>+</sup> sup *<sup>X</sup>*

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

*hs*(*B*) 

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

*Xdμ* =

 0 −∞

*hs*-Measurable sets with finite Hausdorff *s*-dimensional outer measure can be approximated from below by closed subsets [15, p.50] [11, Theorem 1.6 (b)] or equally the restriction of every Hausdorff outer measure *h<sup>s</sup>* to the class of all *hs*-measurable sets with finite Hausdorff outer measure is *inner regular* on the class of all closed subsets of Ω.

In particular any *hs*-measurable set with finite Hausdorff *s*-dimensional outer measure contains an *Fσ*-set of equal measure, and so contains a closed set differing from it by arbitrary small measure.

Since every metric space is a Hausdorff space then every compact subset of Ω is closed; denote by **O** the class of all open sets of Ω and by **C** the class of all compact sets of Ω, the restriction of each Hausdorff *s*-dimensional outer measure to the class **H** of all *hs*-measurable sets with finite Hausdorff outer measure is *strongly regular* [5, p.43] that is it is regular:

a) *<sup>h</sup>s*(*A*) = inf {*hs*(*U*)|*<sup>A</sup>* <sup>⊂</sup> *<sup>U</sup>*, *<sup>U</sup>* <sup>∈</sup> **<sup>O</sup>**} for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>** (outer regular);

b) *<sup>h</sup>s*(*A*) = sup {*hs*(*C*)|*<sup>C</sup>* <sup>⊂</sup> *<sup>A</sup>*, *<sup>C</sup>* <sup>∈</sup> **<sup>C</sup>**} for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>** (inner regular)

with the additional property:

c) inf {*hs*(*<sup>U</sup>* <sup>−</sup> *<sup>A</sup>*)|*<sup>A</sup>* <sup>⊂</sup> *<sup>U</sup>*, *<sup>U</sup>* <sup>∈</sup> **<sup>O</sup>**} <sup>=</sup> 0 for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>**

Any Hausdorff *s*-dimensional outer measure is translation invariant, that is, *hs*(*x* + *E*) = *<sup>h</sup>s*(*E*), where *<sup>x</sup>* <sup>+</sup> *<sup>E</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>* : *<sup>y</sup>* <sup>∈</sup> *<sup>E</sup>*} [11, p.18].

## **3.2. The Choquet integral**

We recall the definition of the Choquet integral [5] with the aim to define upper conditional previsions by Choquet integral with respect to Hausdorff outer measures and to prove their properties. The Choquet integral is an integral with respect to a monotone set function. Given a non-empty set Ω and denoted by *S* a set system, containing the empty set and properly contained in ℘(Ω), the family of all subsets of Ω , a monotone set function *μ*: *S* → �<sup>+</sup> = �<sup>+</sup> ∪ {+∞} is such that *μ*(�) = 0 and if *A*, *B* ∈ *S* with *A* ⊆ *B* then *μ*(*A*) ≤ *μ*(*B*). Given a monotone set function *μ* on *S*, its *outer set function* is the set function *μ*∗ defined on the whole power set ℘(Ω) by

$$\mu^\*(A) = \inf \left\{ \mu(B) : B \supset A; B \in \mathcal{S} \right\}, A \in \wp(\Omega).$$

The inner set function of *μ* is the set function *μ*<sup>∗</sup> defined on the whole power set ℘(Ω) by

$$\mu\_\*(A) = \sup \left\{ \mu(B) | B \subset A; B \in \mathcal{S} \right\}, A \in \wp(\Omega)$$

Let *μ* be a monotone set function defined on *S* properly contained in ℘(Ω) and *X* : Ω → � = � ∪ {−∞, +∞} an arbitrary function on Ω. Then the set function

$$G\_{\mu, \chi}(\mathfrak{x}) = \mu \left\{ \omega \in \Omega : \mathbf{X}(\omega) > \mathfrak{x} \right\}$$

is decreasing and it is called *decreasing distribution function* of *X* with respect to *μ*. If *μ* is continuous from below then *Gμ*,*X*(*x*) is right continuous. In particular the decreasing distribution function of *X* with respect to the Hausdorff outer measures is right continuous since these outer measures are continuous from below. A function *X* : Ω → � is called upper *μ*-measurable if *Gμ*∗,*X*(*x*) = *Gμ*∗,*X*(*x*). Given an upper *μ*-measurable function *X* :Ω → *R* with decreasing distribution function *Gμ*,*X*(*x*), if *μ*(Ω) < +∞, the *asymmetric Choquet integral* of *X* with respect to *μ* is defined by

$$\int \mathcal{X}d\mu = \int\_{-\infty}^{0} (G\_{\mu,X}(\mathfrak{x}) - \mu(\Omega))d\mathfrak{x} + \int\_{0}^{\infty} G\_{\mu,X}(\mathfrak{x})d\mathfrak{x}$$

The integral is in �, can assume the values −∞, +∞ or is undefined when the right-hand side is +∞ − ∞.

If *X* ≥ 0 or *X* ≤ 0 the integral always exists. In particular for *X* ≥ 0 we obtain

$$\int X d\mu = \int\_0^{+\infty} G\_{\mu,X}(x) dx$$

If *X* is bounded and *μ*(Ω) = 1 we have that

$$\int \mathbf{X} d\mu = \int\_{\inf X}^{0} (\mathbf{G}\_{\mu,X}(\mathbf{x}) - 1) d\mathbf{x} + \int\_{0}^{\sup X} \mathbf{G}\_{\mu,X}(\mathbf{x}) d\mathbf{x} = \int\_{\inf X}^{\sup X} \mathbf{G}\_{\mu,X}(\mathbf{x}) d\mathbf{x} + \inf X.$$

#### **3.3. The model**

6 Stochastic Control

lim*i*→<sup>∞</sup> *<sup>h</sup>s*(*Ai*) = *<sup>h</sup>s*(lim*i*→<sup>∞</sup> *Ai*).

*hs*-Measurable sets with finite Hausdorff *s*-dimensional outer measure can be approximated from below by closed subsets [15, p.50] [11, Theorem 1.6 (b)] or equally the restriction of every Hausdorff outer measure *h<sup>s</sup>* to the class of all *hs*-measurable sets with finite Hausdorff outer

In particular any *hs*-measurable set with finite Hausdorff *s*-dimensional outer measure contains an *Fσ*-set of equal measure, and so contains a closed set differing from it by arbitrary

Since every metric space is a Hausdorff space then every compact subset of Ω is closed; denote by **O** the class of all open sets of Ω and by **C** the class of all compact sets of Ω, the restriction of each Hausdorff *s*-dimensional outer measure to the class **H** of all *hs*-measurable sets with

Any Hausdorff *s*-dimensional outer measure is translation invariant, that is, *hs*(*x* + *E*) =

We recall the definition of the Choquet integral [5] with the aim to define upper conditional previsions by Choquet integral with respect to Hausdorff outer measures and to prove their properties. The Choquet integral is an integral with respect to a monotone set function. Given a non-empty set Ω and denoted by *S* a set system, containing the empty set and properly contained in ℘(Ω), the family of all subsets of Ω , a monotone set function *μ*: *S* → �<sup>+</sup> = �<sup>+</sup> ∪ {+∞} is such that *μ*(�) = 0 and if *A*, *B* ∈ *S* with *A* ⊆ *B* then *μ*(*A*) ≤ *μ*(*B*). Given a monotone set function *μ* on *S*, its *outer set function* is the set function *μ*∗ defined on the whole

*μ*∗(*A*) = inf {*μ*(*B*) : *B* ⊃ *A*; *B* ∈ *S*} , *A* ∈ ℘(Ω)

*μ*∗(*A*) = sup {*μ*(*B*)|*B* ⊂ *A*; *B* ∈ *S*} , *A* ∈ ℘(Ω)

Let *μ* be a monotone set function defined on *S* properly contained in ℘(Ω) and *X* : Ω → � =

*Gμ*,*X*(*x*) = *μ* {*ω* ∈ Ω : *X*(*ω*) > *x*}

� ∪ {−∞, +∞} an arbitrary function on Ω. Then the set function

The inner set function of *μ* is the set function *μ*<sup>∗</sup> defined on the whole power set ℘(Ω) by

finite Hausdorff outer measure is *strongly regular* [5, p.43] that is it is regular:

a) *<sup>h</sup>s*(*A*) = inf {*hs*(*U*)|*<sup>A</sup>* <sup>⊂</sup> *<sup>U</sup>*, *<sup>U</sup>* <sup>∈</sup> **<sup>O</sup>**} for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>** (outer regular); b) *<sup>h</sup>s*(*A*) = sup {*hs*(*C*)|*<sup>C</sup>* <sup>⊂</sup> *<sup>A</sup>*, *<sup>C</sup>* <sup>∈</sup> **<sup>C</sup>**} for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>** (inner regular)

c) inf {*hs*(*<sup>U</sup>* <sup>−</sup> *<sup>A</sup>*)|*<sup>A</sup>* <sup>⊂</sup> *<sup>U</sup>*, *<sup>U</sup>* <sup>∈</sup> **<sup>O</sup>**} <sup>=</sup> 0 for all *<sup>A</sup>* <sup>∈</sup> **<sup>H</sup>**

*<sup>h</sup>s*(*E*), where *<sup>x</sup>* <sup>+</sup> *<sup>E</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>* : *<sup>y</sup>* <sup>∈</sup> *<sup>E</sup>*} [11, p.18].

measure is *inner regular* on the class of all closed subsets of Ω.

small measure.

with the additional property:

**3.2. The Choquet integral**

power set ℘(Ω) by

A new model of coherent upper conditional prevision is defined in [9, Theorem 2].

**Theorem 1.** *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 space of bounded random variables on B. Moreover, let m be a 0-1 valued finitely additive, but not countably additive, probability on* ℘(*B*) *such that a different m is chosen for each B. Then for each B* ∈ *B the functional P*(*X*|*B*) *defined on K*(*B*) *by*

 $\overline{P}(X|B) = \frac{1}{h^s(B)}$   $\int\_B X dh^s \text{ if } 0 < h^s(B) < +\infty$ 

*and by*

$$\overline{P}(X|B) = m(XB) \text{ if } h^s(B) = 0, +\infty$$

*is a coherent upper conditional prevision.*

The lower conditional previsions *P*(*A*|*B*) can be define as in the previous theorem if *hs* denotes the Hausdorff s-dimensional inner measure.

Given an upper conditional prevision *P*(*X*|*B*) defined on a linear space the lower conditional prevision *P*(*X*|*B*) is obtained as its conjugate, that is *P*(*X*|*B*) = −*P*(−*X*|*B*). If *B* has positive and finite Hausdorff outer measure in its Hausdorff dimension *s* and we denote by *hs* the Hausdorff *s*-dimensional inner measure we have

$$\underline{P}(X|B) = -\overline{P}(-X|B) = -\frac{1}{h^s(B)} \int\_B (-X)dh^s = 0$$

#### 8 Stochastic Control 58 Stochastic Modeling and Control Coherent Upper Conditional Previsions Defined by Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems <sup>9</sup>

$$\frac{1}{h^s(B)} \int\_B X dh\_s = \frac{1}{h\_s(B)} \int\_B X dh\_s.$$

**Example 3** Let *B* = {*ω*1, *ω*2, ..., *ωn*}. The Hausdorff dimension of *B* is 0 and the Hausdorff 0-dimensional measure *h*<sup>0</sup> is the counting measure. Moreover *h*0(*B*) = *n* then the coherent

A necessary and sufficient condition for an upper prevision *P* to be coherent is to be the *upper envelope* of linear previsions, i.e. there is a class *M* of linear previsions such that *P*

Given a coherent upper prevision *P* defined on a domain **K** the maximal coherent extension of *P* to the class of all bounded random variables is called [19, 3.1.1] *natural extension* of *P*. The linear extension theorem [19, 3.4.2] assures that the class of all linear extensions to the class of all bounded random variables of a linear prevision *P* defined on a linear space **K** is the class *M*(*P*) of all linear previsions that are dominated by *P* on **K**. Moreover the upper and

Let *P*(·|*B*) be the coherent upper conditional prevision on the the class of all bounded

In Doria [9, Theorem 5] it is proven that, for every conditioning event *B*, the given upper conditional prevision is the upper envelope of all linear extensions of *P*(·|*B*) to the class of all

**Theorem 3.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every conditioning event B* ∈ *B let L*(*B*) *be the class of all bounded random variables defined on B and let P*(·|*B*) *be the coherent upper conditional previsionon the class of all bounded Borel-measurable random variables defined in Theorem 1. Then the coherent upper conditional prevision defined on L*(*B*) *as in*

In the same way it can be proven that the conjugate of the coherent upper conditional prevision *P*(·|*B*) is the *lower envelope* of *M*(*P*), the class of all linear extension of *P*(·|*B*)

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

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.

*<sup>B</sup> Xdh<sup>s</sup>* <sup>=</sup> <sup>1</sup>

*<sup>n</sup>* <sup>∑</sup>*<sup>n</sup>*

<sup>1</sup> *X*(*ωi*)

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

Coherent Upper Conditional Previsions De ned by

59

upper conditional prevision is defined for every *X* ∈ **K**(*B*) by

**5. Upper envelope**

=sup{*P*(*X*) : *P* ∈ *M*} [19, 3.3.3].

bounded random variables on *B*.

dominating *P*(·|*B*).

**6. Main results**

associated with *P*(·|*B*).

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

*hs*(*B*) 

lower envelopes of *M*(*P*) are the natural extensions of *P* [19, Corollary 3.4.3].

*Theorem 1 is the upper envelope of all linear extensions of P*(·|*B*) *to the class L*(*B*)*.*

Borel-measurable random variables defined on *B* defined in Theorem 1.

The last equality holds since each *B* is *hs*-measurable, that is *hs*(*B*) = *hs*(*B*).

The unconditional upper prevision is obtained as a particular case when the conditioning event is Ω, that is *P*(*A*) = *P*(*A*|Ω) and *P*(*A*) = *P*(*A*|Ω).

Coherent upper conditional probabilities are obtained when only 0-1 valued random variables are considered; they have been defined in [6] :

**Theorem 2.** *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 m be a 0-1 valued finitely additive, but not countably additive, probability on* ℘(*B*) *such that a different m is chosen for each B. Thus, for each B* ∈ *B, the function defined on* ℘(*B*) *by*

$$
\overline{P}(A|B) = \frac{h^s(AB)}{h^s(B)} \quad \text{if} \quad 0 < h^s(B) < +\infty
$$

*and by*

$$
\overline{P}(A|B) = m(AB) \quad \text{if} \quad \mathbf{h}^s(\mathbf{B}) = 0, +\infty
$$

*is a coherent upper conditional probability.*

Let *B* be a set with positive and finite Hausdorff outer measure in its Hausdorff dimension *s*. Denote by *<sup>h</sup><sup>s</sup>* the *<sup>s</sup>*-dimensional Hausdorff outer measure and for every *<sup>A</sup>* <sup>∈</sup> <sup>℘</sup>(*B*) by *<sup>μ</sup>*<sup>∗</sup> *<sup>B</sup>*(*A*) = *<sup>P</sup>*(*A*|*B*) = *<sup>h</sup><sup>s</sup>*(*AB*) *hs*(*B*) the upper conditional probability defined on ℘(*B*). From Theorem 1 we have that the upper conditional prevision *P*(·|*B*) is a functional defined on **L**(*B*) with values in � and the upper conditional probability *μ*<sup>∗</sup> *<sup>B</sup>* integral represents *P*(*X*|*B*) since *P*(*X*|*B*) = *Xdμ*<sup>∗</sup> *<sup>B</sup>* <sup>=</sup> <sup>1</sup> *hs*(*B*) *Xdhs*. The number <sup>1</sup> *hs*(*B*) is a normalizing constant.

#### **4. Examples**

**Example 1** Let *B* = [0, 1]. The Hausdorff dimension of *B* is 1 and the Hausdorff 1-dimensional measure *h*<sup>1</sup> is the Lebesgue measure. Moreover *h*1[0, 1] = 1 then the coherent upper conditional prevision is defined for every *X* ∈ **K**(*B*) by

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

We recall the definition of the Cantor set. Let *E*<sup>0</sup> = [0,1], *E*<sup>1</sup> = [0,1/3] ∪ [2/3,1], *E*<sup>2</sup> = [0,1/9] ∪ [2/9, 1/3] ∪ [2/3,7/9] ∪ [8/9,1], etc., where *En* is obtained by removing the open middle third of each interval in *En*−1, so *En* is the union of 2*<sup>n</sup>* intervals, each of length <sup>1</sup> <sup>3</sup>*<sup>n</sup>* . The Cantor's set is the perfect set *E* = <sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> *En*.

**Example 2** Let *B* be the Cantor set. The Hausdorff dimension of the Cantor set is *s* = *ln*<sup>2</sup> *ln*<sup>3</sup> and *<sup>h</sup>s*(*B*) = 1. Then the coherent upper conditional prevision is defined for every *<sup>X</sup>* <sup>∈</sup> **<sup>K</sup>**(*B*) by

$$\overline{P}(X|B) = \frac{1}{h^s(B)} \int\_B X dh^s = \int\_B X dh^{\frac{\ln 2}{\ln 3}}$$

**Example 3** Let *B* = {*ω*1, *ω*2, ..., *ωn*}. The Hausdorff dimension of *B* is 0 and the Hausdorff 0-dimensional measure *h*<sup>0</sup> is the counting measure. Moreover *h*0(*B*) = *n* then the coherent upper conditional prevision is defined for every *X* ∈ **K**(*B*) by

 $\overline{P}(X|B) = \frac{1}{h^s(B)}$   $\int\_B X dh^s = \frac{1}{n} \sum\_{l=1}^n X(\omega\_l)$ 
