# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
Fuzzy sets are sets that assign membership degrees or degrees of truth to elements, allowing for a more flexible representation of uncertainty or partial truth. B(I) refers to the Borel \(\sigma\)-algebra, which is the \(\sigma\)-algebra generated by the open sets in the interval I. The Borel \(\sigma\)-algebra consists of all subsets of I that can be formed by taking unions, intersections, and complements of open sets. B(I) \(\otimes\) A represents the product \(\sigma\)-algebra, which is the smallest \(\sigma\)-algebra generated by the sets of the form B \(\times\) A, where B is a Borel set from B(I) and A is a set from the \(\sigma\)-algebra A. The requirement of [X]\({}^{\alpha}\) being B(I) \(\otimes\) A-measurable ensures that the equivalence class [X]\({}^{\alpha}\) is measurable with respect to the joint \(\sigma\)-algebra generated by B(I) and A. This joint \(\sigma\)-algebra allows us to analyse and reason about the measurability properties of the fuzzy stochastic process X across both the time domain (Borel sets in I) and the probability space (sets in A). Suppose that we have two sets: B(I), which represents the Borel \(\sigma\)-algebra generated by the open sets in the interval I, and A, which represents a \(\sigma\)-algebra defined on a sample space \(\Omega\). We want to generate a \(\sigma\)-algebra that captures the joint measurability of sets from B(I) and A. The product \(\sigma\)-algebra B(I) \(\otimes\) A is defined as the smallest \(\sigma\)-algebra generated by sets of the form B \(\times\) A, where B is a Borel set from B(I) and A is a set from the \(\sigma\)-algebra A. Let us consider a specific example to illustrate this concept. Assume that I is the interval [0, 1] and A is the power set of \(\Omega=\{\)a, b\(\}\), which contains all possible subsets of \(\Omega\). B(I): B(I) is the Borel \(\sigma\)-algebra on I, generated by the open sets in I. It includes sets such as open intervals, closed intervals, half-open intervals, and countable unions or intersections of such intervals. A: A is the power set of \(\Omega\), which includes all subsets of \(\Omega\). In this case, A = {\(\emptyset\), \(\{\)a\(\}\), \(\{\)b\(\}\), \(\{\)a\(\}\), b\(\}\)]. To construct the product \(\sigma\)-algebra B(I) \(\otimes\) A, we consider all possible combinations of Borel sets from B(I) and sets from A. For example, let us take B = (0.2, 0.8) (an open interval in I) and A = \(\{\)a\(\}\) (a singleton set in A). The set B \(\times\) A is the Cartesian product of B and A, which results in the set \(\{\)(\(\kappa\), a \(\mid\) 0.2 \(<\)\(\kappa\)\(<\) 0.8\(\}\)]. Similarly, we can consider other combinations of Borel sets from B(I) and sets from A. The product \(\sigma\)-algebra B(I) \(\otimes\) A is then defined as the smallest \(\sigma\)-algebra that includes all such sets B \(\times\) A for all possible choices of B from B(I) and A from A. In this example, the product \(\sigma\)-algebra B(I) \(\otimes\) A would consist of sets such as B \(\times\) A, where B represents Borel sets from B(I) and A represents sets from A. It captures the joint measurability of sets from B(I) and A, allowing us to reason about sets that involve both the real number domain and the sample space. The product \(\sigma\)-algebra B(I) \(\otimes\) A is the smallest \(\sigma\)-algebra that contains all possible combinations of sets formed by pairing a Borel set from B(I) with a set from \(\sigma\)-algebra A. So, B(I) \(\otimes\) A helps us to explore the measurability of events that occur at the intersection of these two domains. A process X is said to be nonanticipating if, for every \(\alpha\in[0,1]\), the multifunction [X]\({}^{\alpha}\) is measurable with respect to the \(\sigma\)-algebra N. The \(\sigma\)-algebra N is defined as follows: \(\mbox{N}:=\{\mbox{A}\in\mbox{B(I)}\otimes\mbox{A}:\mbox{A}^{t}\in\mbox{A}\mbox { for every t}\in\mbox{I}\}\), where A\({}^{t}\) represents the set of outcomes \(\omega\) such that the pair (t, \(\omega\)) belongs to the set A.