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


最后更新: 2023-03-28