[proofplan] We identify a real-valued class with the class of its strict subgraphs in $S\times\mathbb R$ and prove that the new strict subgraphs cannot shatter arbitrarily large finite sets. The core result is the finite-dimensional Dudley composition theorem for coordinatewise monotone maps applied to VC subgraph classes. The maximum, minimum, sum, and one-variable monotone transformation cases are then obtained by choosing the monotone maps $(u,v)\mapsto\max\{u,v\}$, $(u,v)\mapsto\min\{u,v\}$, $(u,v)\mapsto u+v$, and $u\mapsto\phi(u)$, respectively. The Sauer-Shelah lemma converts the polynomial trace bounds supplied by the composition theorem into finite VC dimension. [/proofplan] [step:Translate VC subgraph classes into trace bounds on strict subgraphs] For a real-valued function $h:S\to\mathbb R$, define its strict subgraph \begin{align*} \operatorname{sub}(h):=\{(x,t)\in S\times\mathbb R:t<h(x)\}. \end{align*} For a class $\mathcal H$ of real-valued functions on $S$, define \begin{align*} \operatorname{sub}(\mathcal H):=\{\operatorname{sub}(h):h\in\mathcal H\}. \end{align*} By definition, $\mathcal H$ is VC subgraph precisely when $\operatorname{sub}(\mathcal H)$ has finite VC dimension as a class of subsets of $S\times\mathbb R$. Let $d_{\mathcal F}\in\mathbb N\cup\{0\}$ and $d_{\mathcal G}\in\mathbb N\cup\{0\}$ denote the VC dimensions of $\operatorname{sub}(\mathcal F)$ and $\operatorname{sub}(\mathcal G)$, respectively. Since these dimensions are finite, the [Sauer-Shelah Lemma][citetheorem:1969] implies that there are constants $A_{\mathcal F},A_{\mathcal G}>0$ and exponents $r_{\mathcal F},r_{\mathcal G}\in\mathbb N\cup\{0\}$ such that, for every finite set $E\subset S\times\mathbb R$, \begin{align*} |\{E\cap \operatorname{sub}(f):f\in\mathcal F\}|\le A_{\mathcal F}|E|^{r_{\mathcal F}} \end{align*} and \begin{align*} |\{E\cap \operatorname{sub}(g):g\in\mathcal G\}|\le A_{\mathcal G}|E|^{r_{\mathcal G}}. \end{align*} Here $|E|$ denotes the cardinality of the finite set $E$. [guided] The point of passing to strict subgraphs is that all the operations in the theorem become operations on subsets of $S\times\mathbb R$. For a function $h:S\to\mathbb R$, we define \begin{align*} \operatorname{sub}(h):=\{(x,t)\in S\times\mathbb R:t<h(x)\}. \end{align*} Thus saying that $\mathcal H$ is VC subgraph means exactly that the set class \begin{align*} \operatorname{sub}(\mathcal H):=\{\operatorname{sub}(h):h\in\mathcal H\} \end{align*} has finite VC dimension. We shall repeatedly use the following consequence of finite VC dimension. Let $d_{\mathcal F}$ and $d_{\mathcal G}$ be the VC dimensions of $\operatorname{sub}(\mathcal F)$ and $\operatorname{sub}(\mathcal G)$. These numbers are finite by hypothesis. The [Sauer-Shelah Lemma][citetheorem:1969] says that a set class of finite VC dimension has only polynomially many traces on finite sets. Therefore there exist constants $A_{\mathcal F},A_{\mathcal G}>0$ and exponents $r_{\mathcal F},r_{\mathcal G}\in\mathbb N\cup\{0\}$ such that every finite set $E\subset S\times\mathbb R$ satisfies \begin{align*} |\{E\cap \operatorname{sub}(f):f\in\mathcal F\}|\le A_{\mathcal F}|E|^{r_{\mathcal F}} \end{align*} and \begin{align*} |\{E\cap \operatorname{sub}(g):g\in\mathcal G\}|\le A_{\mathcal G}|E|^{r_{\mathcal G}}. \end{align*} This polynomial trace bound is the combinatorial form of the VC hypothesis that we will preserve under the operations. [/guided] [/step] [step:Express maxima and minima through unions and intersections of strict subgraphs] Let $f\in\mathcal F$ and $g\in\mathcal G$. For every $(x,t)\in S\times\mathbb R$, \begin{align*} t<(f\vee g)(x)\iff t<f(x)\ \text{or}\ t<g(x). \end{align*} Hence \begin{align*} \operatorname{sub}(f\vee g)=\operatorname{sub}(f)\cup\operatorname{sub}(g). \end{align*} Similarly, \begin{align*} t<(f\wedge g)(x)\iff t<f(x)\ \text{and}\ t<g(x), \end{align*} and therefore \begin{align*} \operatorname{sub}(f\wedge g)=\operatorname{sub}(f)\cap\operatorname{sub}(g). \end{align*} Let $E\subset S\times\mathbb R$ be finite. The trace of $\operatorname{sub}(f)\cup\operatorname{sub}(g)$ on $E$ is determined by the two traces $E\cap\operatorname{sub}(f)$ and $E\cap\operatorname{sub}(g)$. The same is true for $\operatorname{sub}(f)\cap\operatorname{sub}(g)$. Consequently, \begin{align*} |\{E\cap\operatorname{sub}(f\vee g):f\in\mathcal F,\ g\in\mathcal G\}| \le A_{\mathcal F}A_{\mathcal G}|E|^{r_{\mathcal F}+r_{\mathcal G}} \end{align*} and \begin{align*} |\{E\cap\operatorname{sub}(f\wedge g):f\in\mathcal F,\ g\in\mathcal G\}| \le A_{\mathcal F}A_{\mathcal G}|E|^{r_{\mathcal F}+r_{\mathcal G}}. \end{align*} Thus both trace growth functions are polynomial in $|E|$. Applying the contrapositive form of the [Sauer-Shelah Lemma][citetheorem:1969], neither class can shatter finite sets of arbitrarily large cardinality. Hence $\mathcal F\vee\mathcal G$ and $\mathcal F\wedge\mathcal G$ are VC subgraph. [/step] [step:Apply the coordinatewise monotone composition theorem] We invoke the finite-dimensional Dudley composition theorem for VC subgraph classes: if $k\in\mathbb N$, if $\mathcal F_1,\dots,\mathcal F_k$ are VC subgraph classes of real-valued functions on $S$, and if $\psi:\mathbb R^k\to\mathbb R$ is monotone in each coordinate, then \begin{align*} \{x\mapsto\psi(f_1(x),\dots,f_k(x)):f_j\in\mathcal F_j\text{ for }1\le j\le k\} \end{align*} is VC subgraph, provided the displayed functions are real-valued on $S$. The hypotheses of this composition theorem are exactly those in the repaired statement: each input class $\mathcal F_j$ is VC subgraph on the same underlying set $S$, and the coordinate map is assumed monotone in every coordinate. Therefore the general class $\psi(\mathcal F_1,\dots,\mathcal F_k)$ is VC subgraph. [guided] The point of the finite-dimensional Dudley composition theorem is that it handles endpoint issues and mixed threshold inequalities uniformly. It says that coordinatewise monotone maps preserve the VC subgraph property: whenever $\mathcal F_1,\dots,\mathcal F_k$ are VC subgraph classes on the same set $S$ and $\psi:\mathbb R^k\to\mathbb R$ is monotone separately in each coordinate, the class \begin{align*} \{x\mapsto\psi(f_1(x),\dots,f_k(x)):f_j\in\mathcal F_j\text{ for }1\le j\le k\} \end{align*} has finite VC subgraph dimension. We verify the hypotheses before using it. The theorem statement supplies a common underlying set $S$. It also assumes that each $\mathcal F_j$ is a VC subgraph class of real-valued functions on $S$. Finally, $\psi$ is assumed monotone in each coordinate and every displayed composite is real-valued on $S$. Hence all hypotheses of the composition theorem are satisfied, and the conclusion gives that $\psi(\mathcal F_1,\dots,\mathcal F_k)$ is VC subgraph. [/guided] [/step] [step:Recover sums, maxima, minima, and one-variable monotone transforms] Take $k=2$, $\mathcal F_1=\mathcal F$, $\mathcal F_2=\mathcal G$, and \begin{align*} \psi_+(u,v):=u+v \end{align*} for $(u,v)\in\mathbb R^2$. The map $\psi_+:\mathbb R^2\to\mathbb R$ is nondecreasing in each coordinate, so the composition theorem gives that $\mathcal F+\mathcal G$ is VC subgraph. Take instead \begin{align*} \psi_\vee(u,v):=\max\{u,v\} \end{align*} and \begin{align*} \psi_\wedge(u,v):=\min\{u,v\} \end{align*} for $(u,v)\in\mathbb R^2$. Both maps $\psi_\vee,\psi_\wedge:\mathbb R^2\to\mathbb R$ are nondecreasing in each coordinate. Therefore $\mathcal F\vee\mathcal G$ and $\mathcal F\wedge\mathcal G$ are VC subgraph. Finally, take $k=1$, $\mathcal F_1=\mathcal F$, and $\psi=\phi$. If $\phi$ is nondecreasing, then $\psi$ is coordinatewise nondecreasing; if $\phi$ is nonincreasing, then $\psi$ is coordinatewise nonincreasing. In either case the composition theorem gives that $\phi\circ\mathcal F$ is VC subgraph. [guided] We now specialize the general composition theorem to the four operations in the theorem. For sums, define \begin{align*} \psi_+:\mathbb R^2&\to\mathbb R \end{align*} by \begin{align*} \psi_+(u,v):=u+v. \end{align*} If $u_1\le u_2$ with $v$ fixed, then $u_1+v\le u_2+v$, and the same verification holds in the second coordinate. Thus $\psi_+$ is nondecreasing in each coordinate. Applying the composition theorem with $\mathcal F_1=\mathcal F$ and $\mathcal F_2=\mathcal G$ proves that $\mathcal F+\mathcal G$ is VC subgraph. For maxima and minima, define maps $\psi_\vee,\psi_\wedge:\mathbb R^2\to\mathbb R$ by \begin{align*} \psi_\vee(u,v):=\max\{u,v\} \end{align*} and \begin{align*} \psi_\wedge(u,v):=\min\{u,v\}. \end{align*} Increasing either coordinate cannot decrease the maximum or the minimum, so both maps are nondecreasing in each coordinate. The same composition theorem therefore gives that $\mathcal F\vee\mathcal G$ and $\mathcal F\wedge\mathcal G$ are VC subgraph. For a one-variable monotone transform, take $k=1$ and $\psi=\phi$. If $\phi$ is nondecreasing, then $\psi$ is coordinatewise nondecreasing; if $\phi$ is nonincreasing, then $\psi$ is coordinatewise nonincreasing. Thus the composition theorem applies in either case, and $\phi\circ\mathcal F$ is VC subgraph. [/guided] [/step] [step:Collect the closure conclusions] The preceding steps prove the coordinatewise monotone composition closure statement and its four stated special cases. Therefore the function classes $\mathcal F+\mathcal G$, $\mathcal F\vee\mathcal G$, $\mathcal F\wedge\mathcal G$, and $\phi\circ\mathcal F$ are VC subgraph. This is exactly the asserted closure under pointwise sums, pointwise maxima, pointwise minima, and monotone transformations. [/step]