# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
These two metrics, d\({}_{\rm s}\) and d\({}_{\infty}\), are used to assess the closeness of fuzzy sets, but they perform this differently. d\({}_{\rm s}\) looks at the behaviour of the fuzzy sets through the continuous \(\lambda\) function and their values at different points. In contrast, d\({}_{\infty}\) considers the worst-case scenario by looking at the maximum difference between the fuzzy set at any point. It is important to note that while being A \(|\,\rm{B}_{do}\)-measurable is more inclusive, the opposite is not necessarily true. Just because X is A \(|\,\rm{B}_{do}\)-measurable does not automatically mean it is A \(|\,\rm{B}_{ds}\)-measurable. Therefore, being A \(|\,\rm{B}_{ds}\)-measurable is a more specific and essential criterion for defining a complete fuzzy random variable. In summary, the given explanations define a fuzzy random variable as a function that assigns fuzzy sets to elements in probability space. It introduces the concept of measurability with respect to different metrics, such as d\({}_{\rm s}\) and d\({}_{\infty}\), and describes the properties and relationships between these metrics and notion of a fuzzy random variable. Next, we expand the work of [16] in a fuzzy random variable and fuzzy stochastic process. A fuzzy random variable X: \(\Omega\to\) F(R) is said to be L\({}^{\rm P}\)-integrally bounded for p \(\geq\) 1 if the equivalence class [X]\({}^{\alpha}\) belongs to the space L\({}^{\rm P}\) (\(\Omega\), A, P; K(R)) for every \(\alpha\in[0,1]\). Here, the random variables X, Y \(\in\) L\({}^{\rm P}\) (\(\Omega\), A, P; K(R)) are identical if P(d\({}_{\infty}\)(X, Y) = 0) = 1. This means that, for X and Y to be considered identical, the probability that their distance d\({}_{\infty}\) is equal to zero should be 1. d\({}_{\infty}\) measures the \"distance\" or dissimilarity between two fuzzy variables X and Y. When the distance d\({}_{\infty}\) between X and Y is zero, it indicates that X and Y are indistinguishable from each other in terms of their fuzzy characteristics. For a fuzzy random variable X: \(\Omega\to\) F(R) and p \(\geq\) 1, the following conditions are equivalent: a) X \(\in\) L\({}^{\rm P}\)(\(\Omega\), A, P; F(R)); b) the equivalence class [X]\({}^{0}\) belongs to L\({}^{\rm P}\)(\(\Omega\), A, P; F(R)); c) the  
that, as t increases, the sub-\(\sigma\)-algebra A\({}_{t}\) becomes larger or contains more sets. In other words, if s \(\leq\) t, then A\({}_{s}\) is a subset of A\({}_{t}\). This property ensures that the information or knowledge available at time s is also available at any later time t \(\geq\) s. The filtration {A\({}_{t}\)}t\(\in\)I is also right-continuous, which means that, for any t in the interval I, the sub-\(\sigma\)-algebra A\({}_{t}\) is equal to the intersection of all sub-\(\sigma\)-algebras A\({}_{s}\) with s \(>\) t. In other words, the information available at time t is already available at any slightly earlier time t \(-\)\(\varepsilon\), where \(\varepsilon\) is a small positive value. The filtration {A\({}_{t}\)}t\(\in\)I is required to contain all P-null sets. A P-null set is a subset of the sample space \(\Omega\) with measure zero. By including all P-null sets in each sub-\(\sigma\)-algebra A\({}_{t}\), the filtration captures all negligible information about the underlying probability space. A fuzzy stochastic process X is said to be d\({}_{\infty}\)-continuous if almost all its trajectories, denoted as X(-, \(\omega\)), are d\({}_{\infty}\)-continuous functions. Here, the trajectories refer to the mappings X(-, \(\omega\)) that associate each time point t in the interval I with a fuzzy set X(t, \(\omega\)) in F(R). The d\({}_{\infty}\)-continuity of these trajectories means that they exhibit continuity with respect to the d\({}_{\infty}\) metric, which measures the dissimilarity between fuzzy sets. A fuzzy stochastic process X is considered as measurable if the equivalence class [X]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R) is a measurable multifunction for all \(\alpha\) in the interval [0, 1].