## 2 Theoretical Framework and Methods
### Uncertainty Measurement Principle
#### 2.2.2 Uncertainty Measurement  
1. Measurable space  
In the domain \(U\), there exists a property domain \(F\), and on the property domain \(F\), there exists a topology \(E\). Then \((F,E)\) is called a measurable space on the domain \(U\) when \(F_{i}\in F(i=1,2,3,\ldots)\) and satisfies Equations (2) and (3). \[F_{i}\cap F_{j}=\varnothing,i\
eq j \tag{2}\]  
\[\underset{i=1}{\cup}F_{i}=F \tag{3}\]  
\(\{F_{1},F_{2},F_{3},\ldots\}\) is a classification of \(F\). Assuming \(\{F_{1},F_{2},F_{3},\ldots,F_{m}\}\) is a partition of \(F\), when the condition (4) is satisfied, \(E\) is called the topology of \(F\). \[E=\left\{E_{i}|E_{i}=\underset{j=1}{\cup}F_{j},F_{j}\in\{\varnothing,F_{1},F_ {2},\ldots,F_{m}\},1\leq i\leq m\right\} \tag{4}\]  
2. Uncertainty measurement  
Assume \((F,E)\) is a measurable space on the domain \(U\), for any \(B\in E\), \(\mu\in U\), there exists a mapping \(u\) such that \(\mu_{B}(\mu)\) satisfies conditions (5)-(7). \[0\leq\mu_{B}(u)\leq 1,\forall u\in U,B\in E \tag{5}\]  
\[\mu_{F}(U)=1 \tag{6}\]  
\[\mu_{\underset{i}{\cup}B_{i}}(u)=\sum_{i}\mu_{B_{i}}(u),B_{i}\in E,B_{i}\cap B _{j}=\varnothing(i\
eq j) \tag{7}\]  
When the three conditions of non-negativity and boundedness (Equation (5)), normalization (Equation (6)), and additivity (Equation (7)) are satisfied, \(\mu_{B}(\mu)\) is referred to as an uncertain measure on the measurable space \((F,E)\). \(\mu_{B}(\mu)\) is the uncertain measure of \(u\) with respect to \(B\), and the uncertain measure essentially represents the degree of distribution of the property \(B_{i}\) in the decision-maker's mind regarding \(u\), which is highly subjective. Assuming there exists a property domain \(F\) on the universe \(U\), and \(E\) is the topology of \(F\), \(\mu_{B}(\mu)\) represents the uncertain measure of \(u\) regarding \(B\), then \((D,E,\mu)\) is referred to as the uncertain measure space. When \(\mu_{B}(\mu)\) is used as the membership function, there exists a fixed \(B\in E\) such that an uncertainty set \(\widetilde{B}\) in the deterministic domain \(U\), then \(\widetilde{B}\) is called an uncertain subset of the universe \(U\) regarding property \(\widetilde{B}\). 3. Uncertainty measurement of Single Indicator  
The evaluation space is the set \(X\) composed of all evaluation influencing factors of the evaluation object \(A\). Let \(\{x_{1},x_{2},\ldots,x_{n}\}\) be the evaluation influencing factors of the evaluation space \(X\), denoted as \(X=\{x_{1},x_{2},\ldots,x_{n}\}\). When there are \(m\) indicators, the influencing factor \(x_{i}(i=1,2,\ldots,m)\) is measured, and the set \(P=\{P_{1},P_{2},\ldots.P_{m}\}\) composed of \(m\) evaluation indicators is called the indicator space. \(x_{ij}\) represents the \(i\)-th evaluation factor for the \(j\)-th indicator \(P_{j}^{\prime}\)'s measurement value, so \(x_{i}\) can be represented as an \(m\) dimensional vector:  
\[x_{i}=(x_{i1},x_{i2},\ldots,x_{im}) \tag{8}\]  
When there are \(Z\) evaluation levels in \(x_{ij}\), the space is denoted as \(C=\{c_{1},c_{2},\ldots,c_{z}\}\), and the \(k\)-th evaluation level is denoted as \(c_{k}\). When the \(k\)-th level is higher than the \(k+1\) level, then \(c_{k}>c_{k+1}(k=1,2,\ldots,z-1)\). When \(c_{1}<c_{2}<\ldots<c_{z}\) or \(c_{1}>c_{2}>\ldots>c_{z}\), then \(\{c_{1},c_{2},\ldots,c_{z}\}\) is referred to as an ordered partition class on the evaluation level space \(C\).