# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
Nonanticipating means that the process \(X\) does not rely on future information but only on the current and past information. By employing the Fubini theorem, the fuzzy integral of a fuzzy stochastic process \(X\), denoted as \(\int_{0}^{T}X(s,\omega)ds\), is defined. Here, \(T\) is a fixed time point, and the integral is taken over the interval from 0 to \(T\). The integral is defined for \(\omega\) in the set \(\Omega\) minus \(N_{X}\), where \(N_{X}\) is a subset of \(\Omega\) belonging to the \(\alpha\)-algebra \(A\) with \(P(N_{X})=0\), meaning that it has measure zero. The fuzzy integral \(\int_{0}^{T}X(s,\omega)ds\) can be defined level-wise. For each \(\alpha\) in the interval [0, 1] and for each \(\omega\) in the set \(\Omega\) minus \(N_{X}\), the Aumann integral \(\int_{0}^{T}[X(s,\omega)]^{a}ds\) belongs to the set K(R), which represents fuzzy sets over the real numbers. Therefore, the fuzzy random variable \(\int_{0}^{T}X(s,\omega)ds\) belongs to the set F(R) for every \(\omega\) in the set \(\Omega\backslash N_{X}\). Now, we define the fuzzy stochastic Lebesgue-Aumann integral of \(X\)\(\epsilon\)\(L^{1}(I\times\Omega\), \(N;\,F(R))\). \[L_{X}(t,w)=\left\{ \begin{matrix}\int_{0}^{T}1_{[0,t]}(s)X(s,w)ds\,\,for\,\,every \,\,w\in\,\,\Omega\backslash N_{x}\\ \langle 0\rangle\,\,for\,\,every\,\,w\,\in\,N_{x}\end{matrix} \right. \tag{8}\]  
We will now discuss properties of this integral \(L_{X}\). First, we introduce some more terminology. The material comes from [25; 28]. The do metric, also known as the supremum or uniform metric, is a mathematical measure of dissimilarity or distance between fuzzy sets. It quantifies the maximum discrepancy between the membership degrees of corresponding elements in two fuzzy sets. Formally, let \(A\) and \(B\) be two fuzzy sets defined on a common universe of discourse. The do metric between \(A\) and \(B\), denoted as \(d\omega(A,\,B)\), is defined as the supremum or maximum of the absolute differences in membership degrees: \(d\omega(A,\,B)=sup\,\,|\mu A(x)-\mu B(x)|\), where \(\mu A(x)\) and \(\mu B(x)\) represent the membership degrees of the element \(x\) in the fuzzy sets \(A\) and \(B\), respectively. The supremum is taken over all elements \(x\) in the universe of discourse. The doe metric provides a measure of the largest difference in membership degrees between corresponding elements of two fuzzy sets. It captures the maximum discrepancy or dissimilarity between the two sets, indicating how much they differ in terms of their membership degrees. Now, we will define the L1 norm. L1 refers to a mathematical space or norm in the context of functions and integrals. Specifically, L1 represents a function space of integrable functions with a finite integral norm. Formally, L1 is a function space defined over a given measure space (\(\Omega\), \(A\), \(\mu\)). It consists of all measurable functions f: \(\Omega\to\mathbb{R}\) (or f: \(\Omega\to\mathbb{C}\)) such that the integral of the absolute value of f, denoted as \(\int|\mathrm{f}|\ \mathrm{d}\mu\), is finite: \(L1(\Omega,\ A,\ \mu)=\big{\{}f:\Omega\to\mathbb{R}(or\ \mathbb{C})\big{|}\big{|}\big{|}f\big{|}\ d\mu<\infty\big{\}}\). In simpler terms, L1 consists of functions whose absolute value is integrable over the measure space, meaning that the area under the curve of the absolute value function is finite. The norm associated with the L1 space, denoted as \(\|\mathrm{f}\|_{1}\), is defined as the integral of the absolute value of f: \(\|\mathrm{f}\|_{1}=\big{\int|\mathrm{f}|\ \mathrm{d}\mu}\). The L1 norm measures the size or magnitude of the function f in terms of its integral, capturing how spread out or concentrated the function is. L1 represents a function space of integrable functions, where the absolute value of the function is integrable. The associated norm, \(\|\mathrm{f}\|_{1}\), measures the size of the function based on its integral. In the context of stochastic processes, the L1 space refers to a function space that consists of processes that are integrable with respect to a given measure.