Harmonizing two approaches to fuzzy random variables

Abstract

We prove a measurability result which implies that the measurable events concerning the values of a fuzzy random variable, in two related mathematical approaches wherein the codomains of the variables are different spaces, are the same (provided both approaches apply). Further results on the perfectness of probability distributions of fuzzy random variables are presented.

1 Introduction

Fuzzy random variables are a well established mathematical model for the simultaneous handling of probabilistic and fuzzy uncertainty. Since fuzziness may appear in almost all aspects of a decision problem, they are not only a tool for statistical data analysis but have also been used in decision making. A few examples are Bayesian decision (Gil and López-Díaz 1996; Rodríguez-Muñiz and López-Díaz 2008), finance problems (Yoshida 2003; Terán 2006; Yoshida et al. 2006), multiobjective decision (Xu and Zeng 2014; Yano 2017), multicriteria decision (Inuiguchi et al. 2016), group decision (Terán and Moreno-Jiménez 2008).

A fuzzy random variable is, intuitively, just a random variable whose values are fuzzy sets instead of numbers. Those values can be given different interpretations (Kruse and Meyer 1987; Couso et al. 2014; Gil et al. 2014), most commonly either as fuzzy perceptions and linguistic descriptions of an underlying precise random variable, or as intrinsically fuzzy data. However, at the mathematical level, fuzzy random variables are significantly harder to handle than ordinary random variables and present specific challenges related to the topology, arithmetics, and order of the fuzzy sets which are allowed as values.

This paper revolves around one such challenge. While random variables take on values in the real line, there are several spaces of fuzzy sets which can serve as the codomain of a fuzzy random variable. Traditionally, for statistical purposes it has been assumed that those fuzzy sets have bounded support (Puri and Ralescu 1986; Kruse and Meyer 1987), and the metrics $d_\infty$ and $d_p$ (Diamond and Kloeden 1994 Chapter 7) had been used. However, Krätschmer (2004) and Krätschmer (2006) and then Trutschnig et al. (2009) and González-Rodríguez et al. (2012) started an elegant approach restricted to convex fuzzy sets but in which some fuzzy sets with unbounded support (and bounded $\alpha$-cuts for $\alpha >0$) are allowed. That space is then endowed with a metric which allows one to embed it as a closed cone into an $L^p$-space (typically $p=2$ since then the final space is a Hilbert space) while being topologically equivalent to $d_p$ in the bounded case.

These approaches are not intrinsically conflictive. The fact that $d_p$ is not complete for fuzzy sets with bounded support means that, if one performed a similar embedding and used known techniques of statistics in Hilbert spaces (if $p=2$) to, e.g., find an estimator, we would risk the situation that this estimator failed to correspond to any fuzzy set in the original space (this is a nice example that topological properties which might appear to be far removed from practice have a definite impact on whether a method will work or not). However, to use this approach one also has to accept some limitations. First, only convex fuzzy sets can be used. Second, while some fuzzy sets with unbounded support are allowed as values, others are not, for no good reason other than mathematical convenience. Moreover, it is not possible to work with the metric $d_\infty$, the strongest in the literature, since it takes infinite values, leading one to settling for weaker consistency and convergence results.

The lack of a single mathematical approach encompassing their advantages makes it important to ensure that both approaches can work in harmony. By this we mean that, if a fuzzy random variable from one framework lives in the space used by the other, we should be able to pass on to that framework and use its results without problems.

Let ${\mathcal{F}_{c}}({\mathbb {R}})$ denote the space of all fuzzy numbers with bounded support (see next section for further details). The question is to model fuzzy random variables as random elements of ${\mathcal{F}_{c}}({\mathbb {R}})$ or of a larger space when the fuzzy values we are working with are actually elements of ${\mathcal{F}_{c}}({\mathbb {R}})$ (which is the case in nearly all applications). These will be called ‘the smaller framework’ and ‘the larger framework’ in the sequel. The ideal situation would be for this mathematical choice to have no working consequence whatsoever.

But an essential question is whether ${\mathcal{F}_{c}}({\mathbb {R}})$ is a measurable set in the larger space (i.e., an element of its $\sigma$-algebra). If that failed, unexpected problems would arise. For example, if ${\mathcal {X}}$ is a fuzzy random variable with distribution $P_{\mathcal {X}}$ and values in ${\mathcal{F}_{c}}({\mathbb {R}})$, then $P_{\mathcal {X}}({\mathcal{F}_{c}}({\mathbb {R}}))=1$ would be a true formula in the smaller framework whereas the quantity $P_{\mathcal {X}}({\mathcal{F}_{c}}({\mathbb {R}}))$ would be undefined in the larger framework.

More generally, the distributions of a fuzzy random variable in both frameworks (the probability measures induced in the two spaces of possible values) would use $\sigma$-algebras with the one for the smaller framework not being contained in the one for the larger framework: the sets of values of which we can speak and calculate probabilities would not be mutually consistent. Another consequence: if ${\mathcal {X}}$ and ${\mathcal {Y}}$ are independent in the smaller framework, by definition

$$\begin{aligned} P_{({\mathcal {X}},{\mathcal {Y}})}(A\times B)=P_{\mathcal {X}}(A)\cdot P_{\mathcal {Y}}(B) \end{aligned}$$

for certain sets of values A, B whereas those probabilities might be undefined in the larger framework as soon as A, B are measurable in ${\mathcal{F}_{c}}({\mathbb {R}})$ but not in the larger space (due to the non-measurability of ${\mathcal{F}_{c}}({\mathbb {R}})$ itself).

The aim of this paper is to prove that, fortunately, these problems actually do not happen as indeed ${\mathcal{F}_{c}}({\mathbb {R}})$ is measurable in the $L^p$-type spaces used in the larger framework. As a subproduct, we also show that all distributions of fuzzy random variables are perfect, a property introduced by Gnedenko and Kolmogorov which avoids some non-intuitive features of arbitrary probability measures.

The structure of the paper is as follows. Next section presents the preliminary notions and results, Sect. 3 proves the measurability result, and Sect. 4 establishes the results about distributions of fuzzy random variables. The paper concludes with some final remarks in Sect. 5.

2 Preliminaries

2.1 Fuzzy sets

Recall a fuzzy subset of ${\mathbb {R}}$ is a function $\widetilde{U}:{\mathbb {R}}\rightarrow [0,1]$. Its $\alpha$-cuts are the sets

$$\begin{aligned} \widetilde{U}_\alpha =\{x\in {\mathbb {R}}\mid \widetilde{U}(x)\ge \alpha \},\quad \alpha \in (0,1], \end{aligned}$$

with the notation $\widetilde{U}_0$ representing the closure of its support.

The set ${\mathcal{F}_{c}}({\mathbb {R}})$ is formed by all fuzzy subsets of ${\mathbb {R}}$ such that $\widetilde{U}_\alpha$ is a non-empty compact interval for all $\alpha \in [0,1]$. The elements of ${\mathcal{F}_{c}}({\mathbb {R}})$ are characterized by the properties of normalization, (fuzzy) convexity and upper semicontinuity.

The metric $d_\infty$ in ${\mathcal{F}_{c}}({\mathbb {R}})$ is defined by

$$\begin{aligned} d_\infty (\widetilde{U},\widetilde{V})=\sup _{\alpha \in (0,1]} d_H(\widetilde{U}_\alpha ,\widetilde{V}_\alpha ), \end{aligned}$$

where $d_H$ is the Hausdorff metric between non-empty compact sets, which in the case of intervals can be written as

$$\begin{aligned} d_H(K,L)=\max \{|\min K-\min L|,|\max K-\max L|\}. \end{aligned}$$

The norm is then the distance to the zero set,

$$\begin{aligned} \Vert K\Vert =d_H(K,\{0\}). \end{aligned}$$

For each $p\in [1,\infty )$, the metric $d_p$ in ${\mathcal{F}_{c}}({\mathbb {R}})$ is defined by

$$\begin{aligned} d_p(\widetilde{U},\widetilde{V})=\left( \int _0^1 d_H(\widetilde{U}_\alpha ,\widetilde{V}_\alpha )^p d\alpha \right) ^{1/p}. \end{aligned}$$

We define now a metric space $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ containing $(\mathcal {F}_c(\mathbb {R}),d_p)$. Set

$$\begin{aligned} \widehat{\mathcal {F}}_{c,p}(\mathbb {R})=\left\{ \widetilde{U}:\mathbb {R} \longrightarrow [0,1] : \forall \alpha \in (0,1]\text { }\widetilde{U}_{\alpha } \in \mathcal {K}_c(\mathbb {R}), \Vert \widetilde{U}\Vert _p < \infty \right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert \widetilde{U}\Vert _p=\left( \int _0^1 \Vert \widetilde{U}_{\alpha }\Vert ^p d \alpha \right) ^{1/p}. \end{aligned}$$

In $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ the definition of the $d_p$-metric still makes sense and, in a natural way, we still use the same notation for it.

2.2 Metric spaces

A Polish space is a topological space whose topology is generated by a complete separable metric. The Borel$\sigma$-algebra$\mathcal {B}_{\mathbb {E}}$ of a metric space ${\mathbb {E}}$ is the smallest $\sigma$-algebra which contains the open sets. In particular, the closed sets and the $G_\delta$-sets (those which can be written as the intersection of countably many open sets) are in $\mathcal {B}_{\mathbb {E}}$.

In the case of $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$, for simplicity the Borel $\sigma$-algebra will be denoted by $\mathcal {B}_{d_p}$.

The space ${\mathcal{F}_{c}}({\mathbb {R}})$ is the one used in the ‘smaller framework’, while some $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ (typically $p=1,2$) is used in the ‘larger framework’. The reader may note, by checking the original sources, that the latter uses similar metrics but not identical to $d_p$. It is known though that those metrics are equivalent (Krätschmer 2004; Trutschnig et al. 2009), whence they generate the same Borel $\sigma$-algebra as $d_p$. Since our proofs involve working with $d_p$, we will not discuss those metrics further here.

The restriction of a function $f:{\mathbb {E}}\rightarrow {\mathbb {R}}$ to a subset $A\subseteq {\mathbb {E}}$ will be denoted by $f|_A$.

2.3 Fuzzy random variables

The notation for a probability space will be $(\Omega ,\mathcal {A},P)$, where $\Omega$ is the sample space, $\mathcal {A}$ the $\sigma$-algebra of events, and P the probability measure.

A mapping $\mathcal {X}:\Omega \rightarrow {\mathcal{F}_{c}}({\mathbb {R}})$ is called a fuzzy random variable if each $\alpha$-set mapping $\mathcal {X}_\alpha :\omega \in \Omega \mapsto (\mathcal {X}(\omega ))_\alpha$ is a random compact interval, equivalently if both $\min \mathcal {X}_\alpha$ and $\max \mathcal {X}_\alpha$ are random variables.

A mapping $\mathcal {X}:\Omega \rightarrow \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ is called a fuzzy random variable if it is measurable with respect to the $\sigma$-algebras $\mathcal {A}$ and $\mathcal {B}_{d_p}$.

That leaves us with two definitions of the same concept, one from each framework. But, as shown by Krätschmer, they are equivalent when both apply.

It is important, however, to underline that the codomain of $\mathcal {X}$ indicates whether $\mathcal {X}$ is being thought of as a fuzzy random variable in one framework or the other.

3 Measurability of ${\mathcal{F}_{c}}({\mathbb {R}})$

In this section, we establish the measurability result which is the basis for the subsequent results in the paper. Our starting point is a continuous extension theorem from Srivastava (1998, Proposition 2.2.3, p. 54).

Lemma 3.1

Let $\mathbb {E}$ be a metric space and $\mathbb {F}$ a complete metric space. Let $A \subseteq \mathbb {E}$. If $f:A \longrightarrow \mathbb {F}$ is continuous, then it has a continuous extension $\bar{f}:B \longrightarrow \mathbb {F}$ to a $G_\delta$-set B which contains A.

That lemma allows us to prove the following support result which has independent interest.

Proposition 3.2

Let $\mathbb {E}$ be a metric space endowed with its Borel $\sigma$-algebra and $f:\mathbb {E} \longrightarrow \mathbb {R}$. If there exists an increasing sequence $\{A_n\}_n$ of (possibly non-measurable) subsets which covers $\mathbb {E}$, and if $f|_{A_n}$ is continuous for each $n \in \mathbb {N}$, then f is a random variable.

Proof

Set $h_n=f|_{A_n}$. For each $n \in \mathbb {N}$, let $B_n$ be the $G_{\delta }$-set provided by Lemma 3.1 such that $A_n \subseteq B_n$, $B_n \in \mathcal {B}_{\mathbb {E}}$, and $\bar{h}_n:B_n \longrightarrow \mathbb {R}$ is a continuous extension of $h_n$ from $A_n$ to $B_n$. Set

$$\begin{aligned} g_n(x)=\left\{ \begin{array}{l@{\quad }l} \bar{h}_n(x) &{} \text {if } x \in B_n,\\ 0 &{} \text {if } x \notin B_n. \end{array} \right. \end{aligned}$$

Let us show each $g_n$ is measurable, namely $g_n^{-1}((-\infty ,a)) \in \mathcal {B}_{\mathbb {E}}$ for each $a \in \mathbb {R}$.

Assume for now $a < 0$. Then

$$\begin{aligned} g_n^{-1}((-\infty ,a))= & {} (B_n \cap g_n^{-1}((-\infty ,a))) \cup (B_n^c \cap g_n^{-1}((-\infty ,a)))\\= & {} \bar{h}_n^{-1}((-\infty ,a)) \cup \emptyset =\bar{h}_n^{-1}((-\infty ,a)) \in \mathcal {B}_{\mathbb {E}} \end{aligned}$$

Else, if $a \ge 0$,

$$\begin{aligned} g_n^{-1}((-\infty ,a))= & {} (B_n \cap g_n^{-1}((-\infty ,a))) \cup (B_n^c \cap g_n^{-1}((-\infty ,a)))\\= & {} \bar{h}_n^{-1}((-\infty ,a)) \cup (B_n^c \cap g_n^{-1}(\{0\}))=\bar{h}_n^{-1}((-\infty ,a)) \cup B_n^c \in \mathcal {B}_{\mathbb {E}} \end{aligned}$$

For each $x \in \mathbb {E}$, there exists $n \in \mathbb {N}$ such that $x \in A_n \subseteq \mathbb {E}$. Therefore, $g_m(x)=f(x)$ for all $m \ge n$, whence f is a random variable because it is the pointwise limit of the measurable functions $g_n$. $\square$

The following left-continuity properties are well known.

Lemma 3.3

Let $\widetilde{U},\widetilde{V} \in \mathcal {F}_c(\mathbb {R})$ and $\alpha \in (0,1]$. If $\alpha _n \nearrow \alpha$, then

• 1. $d_H(\widetilde{U}_{\alpha _n},\widetilde{V}_{\alpha _n}) \rightarrow d_H(\widetilde{U}_{\alpha },\widetilde{V}_{\alpha })$.

• 2. $\Vert \widetilde{U}_{\alpha _n}\Vert \rightarrow \Vert \widetilde{U}_{\alpha }\Vert$.

We will also use the fact that the $d_p$-metrics are increasing in p.

Lemma 3.4

Let $\widetilde{U},\widetilde{V} \in \mathcal {F}_c(\mathbb {R})$. If $1 \le p \le q < \infty$, then $d_p(\widetilde{U},\widetilde{V}) \le d_q(\widetilde{U},\widetilde{V})$.

Proof

Define the function

$$\begin{aligned} f:(0,1]\rightarrow & {} \mathbb {R}\\ \alpha\mapsto & {} d_H(\widetilde{U}_{\alpha },\widetilde{V}_{\alpha }) \end{aligned}$$

By Lemma 3.3, f is left-continuous and thus measurable. Since

$$\begin{aligned} d_p(\widetilde{U},\widetilde{V})=\left( \int _{[0,1]}d_H(\widetilde{U}_{\alpha },\widetilde{V}_{\alpha })^p d\alpha \right) ^{1/p} =\left( \int _{[0,1]}f^pd\alpha \right) ^{1/p}=\Vert f\Vert _p, \end{aligned}$$

applying Minkowski’s inequality we have $\Vert f\Vert _p \le \Vert f\Vert _q$, that is, $d_p(\widetilde{U},\widetilde{V}) \le d_q(\widetilde{U},\widetilde{V})$. $\square$

The main properties of the metric spaces $(\widehat{\mathcal {F}}_{c,p}(\mathbb {R}),d_p)$ were established by Krätschmer (2002, Corollary 3.3).

Lemma 3.5

Let $p\in [1,\infty )$. Then $(\widehat{\mathcal {F}}_{c,p}(\mathbb {R}),d_p)$ is a complete, separable metric space and a completion of $(\mathcal {F}_c(\mathbb {R}),d_p)$ (namely, $\mathcal {F}_c(\mathbb {R})$ is dense in the complete space $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$).

The following real functions play an important role in the proof of our main result. Let $\phi _{\alpha }$ and $\phi _{\alpha ,k}$ be defined as follows.

• For each $\alpha \in (0,1]$,

  $$\begin{aligned} \phi _{\alpha }:\widehat{\mathcal {F}}_{c,p}(\mathbb {R})\rightarrow & {} \mathbb {R}\\ \widetilde{U}\mapsto & {} \Vert \widetilde{U}_{\alpha }\Vert \end{aligned}$$

• For each $k>\frac{1}{\alpha }$,

  $$\begin{aligned} \phi _{\alpha ,k}:\widehat{\mathcal {F}}_{c,p}(\mathbb {R})\rightarrow & {} \mathbb {R}\\ \widetilde{U}\mapsto & {} \left( k\cdot \int _{[\alpha -\frac{1}{k},\alpha ]}\Vert \widetilde{U}_{\beta }\Vert ^p d \beta \right) ^{1/p} \end{aligned}$$

In the sequel, when the values of $\alpha$ and k are fixed by the context, we will denote $I=[\alpha -\frac{1}{k},\alpha ]$.

We will now prove the measurability of the $\phi _{\alpha ,k}$.

Theorem 3.6

The mapping $\phi _{\alpha ,k}$ is a random variable for each $\alpha \in (0,1]$ and $k>\frac{1}{\alpha }$.

Proof

Fix $\alpha \in (0,1],$ and $k>\frac{1}{\alpha }$. Set

$$\begin{aligned} A_n=\left\{ \widetilde{U} \in \widehat{\mathcal {F}}_{c,p}(\mathbb {R}) : \Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert \le n\right\} . \end{aligned}$$

Since $\alpha -\frac{1}{k} >0$, the set $\widetilde{U}_{\alpha -\frac{1}{k}}$ is compact for all $\widetilde{U} \in \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ and then $\Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert <\infty$. Therefore, $\{A_n\}_n$ is an increasing sequence of sets with

$$\begin{aligned} \widehat{\mathcal {F}}_{c,p}(\mathbb {R})=\bigcup _{n \in \mathbb {N}}A_n. \end{aligned}$$

By Proposition 3.2, to complete the proof it is enough to show that the restrictions $\phi _{\alpha ,k}|_{A_n}$ are continuous for all $n \in \mathbb {N}$. We will indeed prove that its power $\phi _{\alpha ,k}^p$ is Lipschitzian in $A_n$, whence, by taking the pth root, the continuity of $\phi _{\alpha ,k}|_{A_n}$ follows.

For any fixed $n \in \mathbb {N}$, let $\widetilde{U},\widetilde{V} \in A_n$. Then

$$\begin{aligned} \Vert \phi _{\alpha ,k}^p(\widetilde{U})-\phi _{\alpha ,k}^p(\widetilde{V})\Vert= & {} \left| k \cdot \int _I \Vert \widetilde{U}_{\beta }\Vert ^p d\beta -k \cdot \int _I \Vert \widetilde{V}_{\beta }\Vert ^p d \beta \right| \nonumber \\\le & {} k \cdot \int _I \left| \Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{V}_{\beta }\Vert ^p \right| d\beta . \end{aligned}$$

(1)

Since $\beta \in I$, we have $\widetilde{U}_{\beta } \subseteq \widetilde{U}_{\alpha -\frac{1}{k}}$, therefore

$$\begin{aligned} 0 \le \Vert \widetilde{U}_{\beta }\Vert \le \Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert \le n. \end{aligned}$$

Analogously, $\Vert \widetilde{V}_{\beta }\Vert \in [0,n]$ as well. Denote by $f_p$ the function $f_p:x \mapsto x^p$; since it is continuously derivable, it is Lipschitzian on [0, n] with the Lipschitz constant

$$\begin{aligned} \sup _{x \in [0,n]}f_p'(x)=\sup _{x \in [0,n]}px^{p-1}=pn^{p-1} \end{aligned}$$

because $p\ge 1$.

Using the triangle inequality for $d_H$,

$$\begin{aligned} \left| \Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{V}_{\beta }\Vert ^p \right|\le & {} pn^{p-1} \left| \Vert \widetilde{U}_{\beta }\Vert -\Vert \widetilde{V}_{\beta }\Vert \right| \\= & {} pn^{p-1}|d_H(\widetilde{U}_{\beta },\{0\})-d_H(\widetilde{V}_{\beta },\{0\})| \le pn^{p-1}d_H(\widetilde{U}_{\beta },\widetilde{V}_{\beta }) \end{aligned}$$

Plugging that into (1),

$$\begin{aligned} \Vert \phi _{\alpha ,k}^p(\widetilde{U})-\phi _{\alpha ,k}^p(\widetilde{V})\Vert\le & {} kpn^{p-1} \int _I d_H(\widetilde{U}_{\beta },\widetilde{V}_{\beta })d\beta \\\le & {} kpn^{n-1} \int _{(0,1]} d_H(\widetilde{U}_{\beta },\widetilde{V}_{\beta })d\beta =kpn^{p-1}d_1(\widetilde{U},\widetilde{V}). \end{aligned}$$

By Lemma 3.4 we know $d_1 \le d_p$, whence

$$\begin{aligned} |\phi _{\alpha ,k}^p(\widetilde{U})-\phi _{\alpha ,k}^p(\widetilde{V})| \le kpn^{p-1}d_p(\widetilde{U},\widetilde{V}). \end{aligned}$$

Indeed $\phi _{\alpha ,k}^p$ is a Lipschitz function on $A_n$ (with constant $kpn^{p-1}$), and the proof is complete. $\square$

We will show now that $\phi _\alpha$ is a random variable.

Proposition 3.7

Let $p \in [1,\infty )$. For each $\widetilde{U} \in \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ and $\alpha \in (0,1]$,

$$\begin{aligned} \phi _{\alpha }(\widetilde{U})=\Vert \widetilde{U}_{\alpha }\Vert =\lim _{k \rightarrow \infty }\phi _{\alpha ,k}(\widetilde{U}). \end{aligned}$$

Proof

Let $\widetilde{U} \in \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$. Since

$$\begin{aligned} \int _I d\beta =\ell (I)=\frac{1}{k} \end{aligned}$$

we have

$$\begin{aligned} \Vert \widetilde{U}_{\alpha }\Vert ^p=\Vert \widetilde{U}_{\alpha }\Vert ^p \cdot k \cdot \int _I d\beta =k\cdot \int _I \Vert \widetilde{U}_{\alpha }\Vert ^p d\beta . \end{aligned}$$

Then

$$\begin{aligned} \left| \phi _{\alpha ,k}^p(\widetilde{U})-\Vert \widetilde{U}_{\alpha }\Vert ^p \right|= & {} \left| k\cdot \int _I \Vert \widetilde{U}_{\beta }\Vert ^pd\beta -k\cdot \int _I \Vert \widetilde{U}_{\alpha }\Vert ^pd\beta \right| \\= & {} \left| k\cdot \int _I (\Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p)d\beta \right| \le k\cdot \int _I \left| \Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p \right| d\beta .\\ | \Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p|= & {} \Vert \widetilde{U}_{\beta }\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p \le \Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p \end{aligned}$$

for all $\beta \in I= [\alpha -\frac{1}{k},\alpha ]$. Therefore

$$\begin{aligned} \left| \phi _{\alpha ,k}^p(\widetilde{U})-\Vert \widetilde{U}_{\alpha }\Vert ^p \right| \le k \cdot \int _I (\Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p)d\beta =k \cdot \frac{1}{k} \cdot (\Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert ^p-\Vert \widetilde{U}_{\alpha }\Vert ^p) \end{aligned}$$

By Lemma 3.3, $\Vert \widetilde{U}_{\alpha -\frac{1}{k}}\Vert \longrightarrow \Vert \widetilde{U}_{\alpha }\Vert$, whence

$$\begin{aligned} \left| \phi _{\alpha ,k}^p(\widetilde{U})-\Vert \widetilde{U}_{\alpha }\Vert ^p \right| \longrightarrow 0 \end{aligned}$$

and then

$$\begin{aligned} \phi _{\alpha ,k}^p(\widetilde{U})\rightarrow \Vert \widetilde{U}_{\alpha }\Vert . \end{aligned}$$

$\square$

Corollary 3.8

Let $\alpha \in (0,1]$. The mapping $\phi _{\alpha }:(\widehat{\mathcal {F}}_{c,p}(\mathbb {R}),\mathcal {B}_{d_p}) \longrightarrow \mathbb {R}$ is a random variable.

Proof

It is so because it is the pointwise limit of the random variables $\phi _{\alpha ,k}$. $\square$

Now we will use the measurability of the $\phi _{\alpha }$ to achieve our main result.

Theorem 3.9

For each $p \in [1,\infty )$, the subspace $\mathcal {F}_c(\mathbb {R})$ is a Borel measurable set in $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$.

Proof

First let us show

$$\begin{aligned} \Vert \widetilde{U}_0\Vert =\sup _n\Vert \widetilde{U}_{1/n}\Vert . \end{aligned}$$

Recall

$$\begin{aligned} \Vert \widetilde{U}_{\alpha }\Vert =\max \{|\inf \widetilde{U}_{\alpha }|,|\sup \widetilde{U}_{\alpha }|\}. \end{aligned}$$

Since $\widetilde{U}_0$ is the closure of $\bigcup _{n=1}^{\infty } \widetilde{U}_{1/n}$, the quantity $|\inf \widetilde{U}_{1/n}|$ decreases to $|\inf \widetilde{U}_0|$ as well as $|\sup \widetilde{U}_{1/n}|$ increases to $|\sup \widetilde{U}_0|$. Thus $\Vert \widetilde{U}_{1/n}\Vert \longrightarrow \Vert \widetilde{U}_0\Vert$ and, being a monotone sequence, $\Vert \widetilde{U}_0\Vert =\sup _n\Vert \widetilde{U}_{1/n}\Vert$.

Now we will show

$$\begin{aligned} \mathcal {F}_c(\mathbb {R})=\bigcup _{n \in \mathbb {N}} \bigcap _{m \in \mathbb {N}} \phi _{1/n}^{-1}(-\infty ,m]. \end{aligned}$$

(2)

$(\subseteq )$ Let $\widetilde{U} \in \mathcal {F}_c(\mathbb {R})$, $\Vert \widetilde{U}_0\Vert <\infty$. Let $m \in \mathbb {N}$ be such that $m\ge \Vert \widetilde{U}_0\Vert$. Then

$$\begin{aligned} \Vert \widetilde{U}_0\Vert&=\sup _n\Vert \widetilde{U}_{1/n}\Vert \Longrightarrow \Vert \widetilde{U}_{1/n}\Vert \le m \text { } \forall n \in \mathbb {N} \Longrightarrow \phi _{1/n}(\widetilde{U}) \le m \ \;\forall n \in \mathbb {N}\\&\Longrightarrow \widetilde{U} \in \phi _{1/n}^{-1}((-\infty ,m]) \;\forall n \in \mathbb {N} \Longrightarrow \widetilde{U} \in \bigcap _{n \in \mathbb {N}} \phi _{1/n}^{-1}((-\infty ,m]). \end{aligned}$$

$(\supseteq )$ Let $\widetilde{U} \in \bigcup _{n \in \mathbb {N}} \bigcap _{m \in \mathbb {N}} \phi _{1/n}^{-1}(-\infty ,m]$. We know $\widetilde{U} \in \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$, there exists $m \in \mathbb {N}$ such that $\phi _{1/n}(\widetilde{U}) \le m$ for all $n \in \mathbb {N}$, yielding $\Vert \widetilde{U}_{1/n}\Vert \le m$ for all $n \in \mathbb {N}$. Then

$$\begin{aligned} \sup _n\Vert \widetilde{U}_{1/n}\Vert \le m \Rightarrow \Vert \widetilde{U}_0\Vert \le m \Rightarrow \Vert \widetilde{U}_0\Vert < \infty \Rightarrow \widetilde{U} \in \mathcal {F}_c(\mathbb {R}). \end{aligned}$$

By Corollary 3.8, $\phi _{\alpha }$ is a random variable whence, for each $n,m\in \mathbb {N}$,

$$\begin{aligned} \phi _{1/n}^{-1}((-\infty ,m]) \in \mathcal {B}_{d_p}. \end{aligned}$$

By (2) and the properties of a $\sigma$-algebra, $\mathcal {F}_c(\mathbb {R}) \in \mathcal {B}_{d_p}$ and the proof is complete. $\square$

4 Distributions of fuzzy random variables with values in ${\mathcal{F}_{c}}({\mathbb {R}})$ and ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$

In this section we use the measurability result Theorem 3.9 to study the distributions of fuzzy random variables.

The following support result is from Parthasarathy (1967, Theorem 1.9).

Lemma 4.1

Let $\mathbb {E}$ be a metric space and let $A \subseteq \mathbb {E}$. If A is a Borel set, then

$$\begin{aligned} \mathcal {B}_A=\{B \in \mathcal {B}_{\mathbb {E}} \mid B \subseteq A\}. \end{aligned}$$

Recall the definition of a fuzzy random variable $\mathcal {X}:\Omega \rightarrow {\mathcal{F}_{c}}({\mathbb {R}})$ as a mapping such that each $\mathcal {X}_\alpha$ is measurable. This is the same as to require $\mathcal {X}$ to be measurable with respect to the $\sigma$-algebras $\mathcal {A}$ and $\sigma _L$, the smallest $\sigma$-algebra which makes the mappings $L_\alpha :\widetilde{U}\in {\mathcal{F}_{c}}({\mathbb {R}})\mapsto \widetilde{U}_\alpha$ measurable.

Accordingly, the distribution of a fuzzy random variable $\mathcal {X}$ in the ‘smaller framework’ is a probability measure on the $\sigma$-algebra $\sigma _L$ of subsets of ${\mathcal{F}_{c}}({\mathbb {R}})$ whereas its distribution in the ‘larger framework’ is a probability measure on $\sigma _{d_p}$, a family of subsets of ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$.

Theorem 4.2

The $\sigma$-algebras associated to the distributions of ${\mathcal{F}_{c}}({\mathbb {R}})$-valued and ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$-valued fuzzy random variables satisfy the following.

• (a) If $A\in \sigma _L$ then $A\in \sigma _{d_p}$.

• (b) If $A\in \sigma _{d_p}$ and $A\subset {\mathcal{F}_{c}}({\mathbb {R}})$, then $A\in \sigma _L$.

Proof

By part (ii)$\Leftrightarrow$(iv) in Krätschmer (2001, Theorem 6.6), $\sigma _L$ is the Borel $\sigma$-algebra of ${\mathcal{F}_{c}}({\mathbb {R}})$ endowed with the $d_p$ metric. By Lemma 4.1, it is formed by the elements of $\mathcal {B}_{d_p}$ which are contained in ${\mathcal{F}_{c}}({\mathbb {R}})$. That proves part (a) as well as part (b). $\square$

Theorem 4.2 means that, whenever $\mathcal {X}$ is a fuzzy random variable with values in ${\mathcal{F}_{c}}({\mathbb {R}})$, it can be handled in both frameworks and the events $A\subset {\mathcal{F}_{c}}({\mathbb {R}})$ of possible values for which the expression $P_{\mathcal {X}}(A)$ makes sense are exactly the same.

In the remainder of this section, we show that all distributions of fuzzy random variables, with values in either ${\mathcal{F}_{c}}({\mathbb {R}})$ or ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$, are perfect.

Kolmogorov’s approach to probability, now standard, is based on measure theory and thus the sample space is allowed to be an arbitrary set. Former approaches, like e.g. Paul Lévy’s, constructed probability distributions as limits of discrete (finite) situations and thus would not allow for arbitrary sample spaces (one may remark that still today explicit constructions from finite situations are favoured by some, e.g. Jaynes 2003).

However, Kolmogorov’s generality gave rise in the 1940s to a number of striking or pathological examples. One such situation is the following, observed by Doob (1948) and Jessen (1948).

Proposition 4.3

(Doob–Jessen) There exists a probability space $(\Omega ,\mathcal {A},P)$, independent random variables $\xi ,\eta :\Omega \longrightarrow \mathbb {R}$ and sets $A,B \subseteq \mathbb {R}$ such that

• (a) $\{{\xi } \in A\} \in \mathcal {A}$, $\{{\eta } \in B\} \in \mathcal {A}.$

• (b) $P({\xi } \in A, {\eta } \in B) \ne P({\xi } \in A) \cdot P({\eta } \in B)$.

At first sight this seems self-contradictory; the explanation is that such A, B are not Borel sets whereas distributions of random variables are defined in the $\sigma$-algebra of Borel sets. Thus ${\xi },{\eta }$ can satisfy the formal definition of independence while they are intuitively not independent since

$$\begin{aligned} P({\xi }\in A\mid {\eta }\in B)\not =P({\xi }\in A) \end{aligned}$$

for certain sets A, B.

To overcome this and other anti-intuitive situations (see, e.g., Blackwell 1956), the notions of a perfect probability measure and a perfect measurable space were introduced by Gnedenko and Kolmogorov (1954). A probability measure P in a measurable space $(\Omega ,\mathcal {A})$ is called perfect if, for every $A \subseteq \mathbb {R}$ and every random variable ${\xi }:\Omega \longrightarrow \mathbb {R}$ such that $\{{\xi } \in A\} \in \mathcal {A}$, there exist Borel sets $B_1,B_2 \subseteq {\mathbb {R}}$ such that

• $B_1 \subseteq A \subseteq B_2$.

• $P({\xi } \in B_2 {\setminus } B_1)=0.$

Thus a perfect probability measure is a probability measure which leaves no room for a situation like Proposition 4.3 to happen since all the information about the values of ${\xi }$ in the experiment is contained in the Borel sets: if A is non-Borel then, for appropriate Borel sets $B_1\subseteq A\subseteq B_2$, we have

$$\begin{aligned} P({\xi }\in A)=P({\xi }\in B_1)=P({\xi }\in B_2). \end{aligned}$$

In its turn, a measurable space is called perfect if all probability measures which can be defined on it are perfect.

Let us show that all distributions of fuzzy random variables with values in ${\mathcal{F}_{c}}({\mathbb {R}})$ or ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$ are perfect.

Lemma 4.4

Every Polish space, endowed with its Borel $\sigma$-algebra, is perfect.

Proof

This follows from Ramachandran (2002, Facts C1 and P2, pp. 770–771). Namely, every probability measure in a Polish space is compact in the sense of measure theory, and every compact probability measure is perfect. $\square$

It is immediate that ${\widehat{\mathcal {F}}_{c,p}}({\mathbb {R}})$ is a perfect space, since it is Polish because the metric $d_p$ is separable and complete.

Corollary 4.5

Let $p\in [1,\infty )$. Let $\mathcal {{\mathcal {X}}}:\Omega \longrightarrow \widehat{\mathcal {F}}_{c,p}(\mathbb {R})$ be a fuzzy random variable. Then its distribution $P_{\mathcal {X}}$ is perfect.

To show that ${\mathcal{F}_{c}}({\mathbb {R}})$ is perfect too we need another known result (Fremlin 2002, Exercise 342X.(n).(i), p. 181).

Lemma 4.6

Every measurable subspace of a perfect measurable space is perfect.

Therefore distributions in ${\mathcal{F}_{c}}({\mathbb {R}})$ are perfect.

Proposition 4.7

Let $\mathcal {X}:\Omega \longrightarrow \mathcal {F}_c(\mathbb {R})$ be a fuzzy random variable. Then its distribution $P_{\mathcal {X}}$ is perfect.

Proof

By Theorem 3.9, $\mathcal {F}_c(\mathbb {R})$ is a measurable subspace of the perfect measurable space $\widehat{\mathcal {F}}_{c,p}(\mathbb {R})$. $\square$

5 Concluding remarks

The informal terms ‘smaller framework’ and ‘larger framework’ only make sense in the context of variables with convex values, since the latter stops to apply in the case of non-convex values.

We have dealt with fuzzy random variables with convex values in ${\mathcal{F}_{c}}({\mathbb {R}})$. It should be emphasized that a similar concern with the relationship between ${\mathcal{F}_{c}}({\mathbb {R}})$ and its superspace ${\mathcal{F}}({\mathbb {R}})$ formed by withdrawing the requirement of convexity is not necessary, as it is not hard to show that ${\mathcal{F}_{c}}({\mathbb {R}})$ is measurable in ${\mathcal{F}}({\mathbb {R}})$.

Indeed, the space of all compact convex subsets is closed in the space of all compact subsets (e.g. Li et al. 2002, Theorem 1.1.2), whence each set

$$\begin{aligned} \{\widetilde{U}\in {\mathcal{F}}({\mathbb {R}})\mid \widetilde{U}_\alpha \hbox { is convex}\} \end{aligned}$$

is measurable. Writing ${\mathcal{F}_{c}}({\mathbb {R}})$ as the countable intersection

$$\begin{aligned} {\mathcal{F}_{c}}({\mathbb {R}})=\bigcap _{\alpha \in (0,1]\cap \mathbb {Q}}\{\widetilde{U}\in {\mathcal{F}}({\mathbb {R}})\mid \widetilde{U}_\alpha \hbox { is convex}\}, \end{aligned}$$

we see that it is indeed measurable.