# 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 summary, a \(\sigma\)-algebra of subsets of \(\Omega\) is a collection of subsets that satisfies specific properties, including containing the empty set, being closed under complementation, and being closed under countable unions. It provides a foundation for defining measurable sets and functions, allowing us to study measures and perform various mathematical operations on these sets. A fuzzy random variable is defined as a function X: \(\Omega\to\) F(R), where F(R) represents the set of fuzzy sets over the real numbers. This means that X assigns each element \(\omega\) in \(\Omega\) a fuzzy set X(\(\omega\)) in F(R). For every \(\alpha\in[0,1]\), the mapping \([\mathrm{X}]^{\alpha}\): \(\Omega\to\) K(R) (where K(R) denotes the set of fuzzy sets over R) should be A-measurable. This implies that the pre-images of measurable sets in K(R) under \([\mathrm{X}]^{\alpha}\) should be measurable sets in A. Consider a metric \(\rho\) on the set F(R) and let B\({}_{\rho}\) be the \(\sigma\)-algebra generated by the topology induced by \(\rho\). A fuzzy random variable X can be seen as a measurable mapping between the measurable spaces (\(\Omega\), A) and (F(R), B\({}_{\rho}\)), and we refer to X as being A\(\mid\)B\(\rho\)-measurable. The metric d\({}_{\mathrm{s}}\)(u, v) is defined as follows: d\({}_{\mathrm{s}}\)(u, v) := inf max {sup \(\mid\)\(\lambda\)(t) - t\(\mid\), sup d\({}_{\mathrm{H}}\)(X\({}_{\mathrm{u}}\)(t), X\({}_{\mathrm{u}}\)(\(\lambda\)(t))), where \(\lambda\)\(\in\)\(\Phi\), t \(\in[0,1]\) and t\(\in[0,1]\), \(\Phi\) denotes the set of strictly increasing continuous functions \(\lambda\): \([0,1]\to[0,1]\) satisfying \(\lambda(0)=0\) and \(\lambda(1)=1\). X\({}_{\mathrm{u}}\) and X\({}_{\mathrm{v}}\) are cadlag (right-continuous with left limits) representations of the fuzzy sets u and v in F(R), respectively. d\({}_{\mathrm{H}}\) represents the Hausdorff distance between fuzzy sets. The space (F(R), d\({}_{\infty}\)) is complete (meaning that all Cauchy sequences converge) and non-separable (does not contain a countable dense subset), while the space (F(R), d\({}_{\mathrm{s}}\)) is a Polish metric space (a complete and separable metric space). The reason for the difference in these properties lies in their respective definitions and metrics. The metric d\(\infty\), defined as the supremum of the Hausdorff distances between equivalence classes [u]\({}^{\alpha}\) and [v]\({}^{\alpha}\), is a robust way to measure the \"closeness\" of fuzzy sets. It can be shown that the space (F(R), d\({}_{\infty}\)) is complete, meaning all Cauchy sequences converge. However, d\(\infty\) does not lend itself to finding a countable dense subset within the space (F(R), d\({}_{\infty}\)). This makes (F(R), d\({}_{\infty}\)) non-separable because it lacks the property of separability, even though it is complete. On the other hand, the metric d\({}_{\mathrm{s}}\) is defined differently and results in a metric space that is both complete and separable. This metric is designed in such a way that it allows for the existence of a countable subset within the space (F(R), d\({}_{\rm s}\)). Because (F(R), d\({}_{\rm s}\)) is complete (all Cauchy sequences converge) and separable (contains a countable dense subset), it meets the criteria to be classified as a Polish metric space. For a mapping X: \(\Omega\to\) F(R) on the probability space (\(\Omega\), A, P), X is a fuzzy random variable iff X is A \(|\,\rm{B}_{ds}\)-measurable and, if X is A \(|\,\rm{B}_{do}\)-measurable, then it is a fuzzy random variable, but the opposite is not necessarily true. For X to be considered a fuzzy random variable, it must meet the condition of being A \(|\,\rm{B}_{ds}\)-measurable. This means that X can be measured using the d\({}_{\rm s}\) metric. The d\({}_{\rm s}\) metric is a specific way to compare how close or similar two fuzzy sets are. If X is also A \(|\,\rm{B}_{do}\)-measurable, it indicates that X can be analysed using the do metric.