Theorem 1. Suppose $$f: \mathcal{X} \rightarrow \mathbb{R}$$ is a continuous set function w.r.t Hausdorff distance $$d_{H}(\cdot, \cdot) . \quad \forall \epsilon>$$ $$0, \exists$$ a continuous function $$h$$ and a symmetric function $$g\left(x_{1}, \ldots, x_{n}\right)=\gamma \circ M A X$$, such that for any $$S \in \mathcal{X}$$,

$\left|f(S)-\gamma\left(\operatorname{MAX}_{x_{i} \in S}\left\{h\left(x_{i}\right)\right\}\right)\right|<\epsilon$

Theorem 2. Suppose $$\mathbf{u}: \mathcal{X} \rightarrow \mathbb{R}^{K}$$ such that $$\mathbf{u}=$$ $$\underset{x_{i} \in S}{\operatorname{MAX}}\left\{h\left(x_{i}\right)\right\}$$ and $$f=\gamma \circ \mathbf{u}$$. Then,

(a) $$\forall S, \exists \mathcal{C}_{S}, \mathcal{N}_{S} \subseteq \mathcal{X}, f(T)=f(S)$$ if $$\mathcal{C}_{S} \subseteq T \subseteq \mathcal{N}_{S} ;$$

(b) $$\left|\mathcal{C}_{S}\right| \leq K$$