# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
By defining the metric d\({}_{\infty}\) this way, we establish that F(R) is a complete metric space. Completeness means that every Cauchy sequence in F(R) converges to a limit within F(R). In other words, F(R) with the metric d\({}_{\infty}\) satisfies the property that any sequence of functions that gets arbitrarily close to each other eventually has a limit within the same set. So, we have defined the metric d\({}_{\infty}\) on the set of real valued functions F(R) and established F(R) equipped with d\({}_{\infty}\) as a complete metric space. The metric measures the Hausdorff distance between equivalence classes of functions under a parameter \(a\) in the interval [0, 1]. For every u, v, w, z \(\epsilon\) F(R), \(\lambda\)\(\epsilon\) R, we have the following properties:  
1. Additive property d\({}_{\infty}\)(u + w, v + w) = d\({}_{\infty}\)(u,v);
2. Distribution property d\({}_{\infty}\)(u + v, w + z) = d\({}_{\infty}\)(u,w) + d\({}_{\infty}\)(v,z);
3. Transitive (triangular) property d\({}_{\infty}\)(u,v) \(\leq\) d\({}_{\infty}\)(u,w) + d\({}_{\infty}\)(w,v);
4. Scalar multiplication property d\({}_{\infty}\)(\(\lambda\)u, \(\lambda\)v) = \(\mid\)\(\lambda\)\(\mid\) d\({}_{\infty}\)(u,v);
5. Non-negativity d\({}_{\infty}\)(u, v) is non-negative for all u, v in F(R);
6. Identity of indiscernibles d\({}_{\infty}\)(u, v) = 0 if and only if u = v;
7. Symmetry d\({}_{\infty}\)(u, v) = d\({}_{\infty}\)(v, u) for all u, v in F(R);
8. Consistency d\({}_{\infty}\)(u, v) = d\({}_{\infty}\)(v, u) = 0 implies u = v.  
Expanding the work of [16], we will now define a fuzzy random variable as a function that assigns fuzzy sets to elements in probability space. We will introduce the concept of measurability with respect to different metrics such as d\({}_{\infty}\) and d\({}_{\rm s}\) and describe the properties and relationships between these metrics and notion of a fuzzy random variable. We will break it down step by step:  
Let (\(\Omega\), A, P) be a probability space, where \(\Omega\) represents the sample space, A is a \(\sigma\)-algebra of subsets of \(\Omega\), and P is a probability measure defined on A. Given a set \(\Omega\), a \(\sigma\)-algebra of subsets of \(\Omega\), denoted as \(\Sigma\), is a collection of subsets of \(\Omega\) that satisfies certain properties. It is a fundamental concept in measure theory and serves as a foundation for defining measurable sets and functions. To formally define a \(\sigma\)-algebra, we require it to satisfy the following properties:  
* The empty set (\(\mathcal{O}\)) is in \(\Sigma\): \(\mathcal{O}\in\Sigma\). * \(\Sigma\) is closed under complementation: if A is in \(\Sigma\), then its complement, denoted as A\({}^{c}\), is also in \(\Sigma\). * \(\Sigma\) is closed under countable unions: if A\({}_{1}\), A\({}_{2}\), A\({}_{3}\), \(\ldots\) are subsets of \(\Omega\) and each A\({}_{i}\) is in \(\Sigma\), then the union of all the A\({}_{i}\), denoted as \(\bigcup\) A\({}_{i}\), is also in \(\Sigma\). In other words, a \(\sigma\)-algebra is a collection of subsets of \(\Omega\) that includes the empty set, is closed under the taking of complements, and is closed under countable unions. The purpose of defining a \(\sigma\)-algebra is to identify a set of subsets of \(\Omega\) that can be considered \"measurable\" in a certain sense. The elements of \(\Sigma\) are often referred to as measurable sets, and they form a structure that allows us to define measures (such as probability measures) on these sets. By specifying a \(\sigma\)-algebra on \(\Omega\), we establish a framework for defining and working with measurable functions, integrating functions, and constructing measures. The \(\sigma\)-algebra provides a systematic way to determine which subsets of \(\Omega\) are considered \"measurable\" within the given context, allowing us to analyse and reason about these subsets in a rigorous mathematical manner.