# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
In other words, N consists of subsets of I \(\times\)\(\Omega\) for which the corresponding time t belongs to the set A. Before proceeding further, we need to state a few theorems. Definition 1: The [PERSON]'s theorem [26], also known as the Fubini-Tonelli theorem, is a fundamental result in measure theory and integration theory. It provides a condition under which the iterated integrals of a measurable function over a product space can be computed by performing separate integrations. Formally, let (\(\Omega_{1}\), \(A_{1}\), \(\mu_{1}\)) and (\(\Omega_{2}\), \(A_{2}\), \(\mu_{2}\)) be two measure spaces, where \(\Omega_{1}\) and \(\Omega_{2}\) are sets, \(A_{1}\) and \(A_{2}\) are \(\sigma\)-algebras of subsets of \(\Omega_{1}\) and \(\Omega_{2}\), respectively, and \(\mu_{1}\) and \(\mu_{2}\) are measures defined on \(A_{1}\) and \(A_{2}\), respectively. Consider the product measure space (\(\Omega\), \(A\), \(\mu\)) defined as the product of the two measure spaces, where \(\Omega=\Omega_{1}\)\(\times\)\(\Omega_{2}\), \(A=A_{1}\)\(\otimes\)\(A_{2}\) (the product \(\sigma\)-algebra), and \(\mu\) is the product measure. [PERSON]'s theorem states that if: \(\Omega\)\(\rightarrow\) [0, +\(\infty\)] is a non-negative measurable function, then the iterated integrals \(\int\Omega_{1}\int\Omega_{2}\)\(f(\omega_{1}\), \(\omega_{2})\)\(d\mu_{1}(\omega_{1})\)\(d\mu_{2}(\omega_{2})\) and \(\int\Omega_{2}\int\Omega_{1}\)\(f(\omega_{1}\), \(\omega_{2})\)\(d\mu_{2}(\omega_{2})\)\(d\mu_{1}(\omega_{1})\) are both well defined and equal, provided at least one of them is finite. In other words, under appropriate conditions, the order of integration can be interchanged without affecting the result. **Definition 2**. : _The Aumann integral, also known as the integral with respect to a fuzzy measure, is a generalisation of traditional integration theory that applies to fuzzy sets to extend the concept of integration to accommodate uncertainty and partial truth [27]. Traditional integration measures the \"area under the curve\" or the accumulation of quantity over a given range. With fuzzy sets, where membership degrees represent degrees of truth or uncertainty, the concept of area or accumulation needs to be generalised. The Aumann integral achieves this by integrating a fuzzy set with respect to a fuzzy measure, which assigns a degree of \"importance\" or \"size\" to subsets of a set, similar to how a traditional measure assigns a value to subsets. Formally, let \(X\) be a fuzzy set defined on a set \(\Omega\) and let \(\mu\) be a fuzzy measure defined on the power set of \(\Omega\). The Aumann integral of \(X\) with respect to \(\mu\), denoted as \(\int\,X\,d\mu\), is a generalised notion of integration that takes into account the degrees of truth or membership of \(X\). The Aumann integral is typically defined in terms of a level-wise integral. For each \(\alpha\) in the interval [0, 1], the Aumann integral \(\int\,X\,d\mu\) is equal to the integral of the \(\alpha\)-cut of \(X\) (i.e., the subset of elements with a membership degree of at least \(\alpha\)) with respect to the \(\alpha\)-cut of the fuzzy measure \(\mu\). The Aumann integral allows for the accumulation and aggregation of fuzzy information._  
Following the works of [16], we will now use [PERSON]'s theorem and Aumann integral to build the fuzzy integral and discuss its properties. A fuzzy stochastic process \(X\) is called \(L^{P}\)-integrably bounded (\(p\geq 1\)) if there exists a real-valued stochastic process \(h\in L^{P}(I\times\Omega\), \(N;R+)\), where \(I\) is the interval and \(\Omega\) is the probability space, such that the fuzzy set \(\mid\mid X(t,\,\omega)\rangle^{0}\mid\mid\) is bounded by \(h(t,\,\omega)\) for almost all (\(t,\,\omega\)) in \(I\times\Omega\). The notation \(\mid\mid\mid X(t,\,\omega)\rangle^{0}\mid\mid\) represents the magnitude or absolute value of the equivalence class \([X(t,\,\omega)]^{0}\). \(L^{P}(I\times\Omega\), \(N;\,F(R))\) denotes the set of nonanticipating and \(L^{P}\)-integrably bounded fuzzy stochastic processes.