39. Theory of Maximum

Jun He June 01, 2026

中文English 最大值定理Theory of Maximum 上半连续Upper Hemi-continuous 紧值对应Compact-valued 数学附录Math Appendix

39. 最大值定理

Important

定理 39.1（最大值定理）对 $X\subseteq\mathbb{R}^n$ 与 $Y\subseteq\mathbb{R}^m$，若 $f:X\times Y\to\mathbb{R}$ 关于 $x$ 与 $y$ 连续，$\Gamma:X\to Y$ 紧值（compact-valued）且连续，则由 $$ > h(x)=\max_{y\in\Gamma(x)}f(x,y) > $$ 定义的函数 $h:X\to\mathbb{R}$ 连续；且由 $G(x)=\{y\in\Gamma(x):f(x,y)=h(x)\}$ 定义的对应 $G:X\to Y$ 非空、紧值、上半连续（upper hemi-continuous）。

这里 $f(x,y)$ 即 $F(x,y)+\beta\tilde v(y)$，$h(x)$ 即 $(T\tilde v)(x)$。（即把它用于动态规划中的贝尔曼算子 $T$：$\tilde v$ 连续、$\Gamma$ 紧值连续 ⟹ $T\tilde v$ 连续，从而 $T$ 把连续函数映为连续函数。）

39. Theory of Maximum

Important

Theorem 39.1 (Theory of Maximum) For $X\subseteq\mathbb{R}^n$ and $Y\subseteq\mathbb{R}^m$, if $f:X\times Y\to\mathbb{R}$ is continuous in $x$ and $y$, $\Gamma:X\to Y$ is compact-valued and continuous, then the function $h:X\to\mathbb{R}$ defined by $$ > h(x)=\max_{y\in\Gamma(x)}f(x,y) > $$ is continuous; and the correspondence $G:X\to Y$ defined by $G(x)=\{y\in\Gamma(x):f(x,y)=h(x)\}$ is non-empty, compact-valued and upper hemi-continuous.

Here $f(x,y)$ is $F(x,y)+\beta\tilde v(y)$ and $h(x)$ is $(T\tilde v)(x)$. (That is, applied to the Bellman operator $T$ in dynamic programming: $\tilde v$ continuous and $\Gamma$ compact-valued and continuous imply $T\tilde v$ is continuous, so that $T$ maps continuous functions to continuous functions.)