[proofplan] The proof is by the bracketing criterion for Donsker classes. First we show that the Lipschitz condition gives a common $L^2(P)$ envelope, so every function in the class lies in $L^2(P)$. Then a finite Euclidean net of the compact parameter set produces $L^2(P)$-brackets whose widths are controlled by the $L^2(P)$ norm of $M$. Compactness of $\Theta\subset\mathbb R^d$ gives polynomial covering growth, so the bracketing entropy integral is finite, and the [bracketing Donsker theorem](/theorems/9836) applies. [/proofplan]

[step:Replace the Lipschitz factor by a nonnegative $L^2(P)$ envelope] Define the [measurable function](/page/Measurable%20Function) \begin{align*} L:E\to[0,\infty) \end{align*} by \begin{align*} L(x):=|M(x)|. \end{align*} Since $M\in L^2(E,\mathcal E,P)$, also $L\in L^2(E,\mathcal E,P)$ and \begin{align*} \|L\|_{L^2(P)}=\|M\|_{L^2(P)}. \end{align*} For every $x\in E\setminus N$ and every $\theta,\eta\in\Theta$, the stated Lipschitz hypothesis implies \begin{align*} |f_\theta(x)-f_\eta(x)|\le L(x)|\theta-\eta|. \end{align*} Indeed, the assumed inequality gives a nonnegative left-hand side bounded above by $M(x)|\theta-\eta|$, and since $M(x)\le |M(x)|$ while $|\theta-\eta|\ge 0$, it also gives the weaker bound with $|M(x)|$ in place of $M(x)$. [/step]

[step:Show that every function in the class is square-integrable] Let \begin{align*} D_\Theta:=\sup_{\theta,\eta\in\Theta}|\theta-\eta| \end{align*} denote the Euclidean diameter of $\Theta$. Since $\Theta$ is compact in $\mathbb R^d$, $D_\Theta<\infty$. For every $\theta\in\Theta$ and every $x\in E\setminus N$, \begin{align*} |f_\theta(x)|\le |f_{\theta_*}(x)|+|f_\theta(x)-f_{\theta_*}(x)| \end{align*} by the triangle inequality. Applying the Lipschitz bound with $\eta=\theta_*$ gives \begin{align*} |f_\theta(x)|\le |f_{\theta_*}(x)|+D_\Theta L(x). \end{align*} Define the measurable function \begin{align*} F:E\to[0,\infty) \end{align*} by \begin{align*} F(x):=|f_{\theta_*}(x)|+D_\Theta L(x). \end{align*} The function $F$ belongs to $L^2(E,\mathcal E,P)$ because $f_{\theta_*}\in L^2(E,\mathcal E,P)$, $L\in L^2(E,\mathcal E,P)$, and $D_\Theta<\infty$. The preceding inequality shows that $|f_\theta(x)|\le F(x)$ for every $\theta\in\Theta$ and every $x\in E\setminus N$. Therefore $F$ is a measurable square-integrable envelope for $\mathcal F$, and in particular $f_\theta\in L^2(E,\mathcal E,P)$ for every $\theta\in\Theta$. [/step]

[step:Build $L^2(P)$ brackets from a Euclidean net of $\Theta$]Fix $\varepsilon>0$. If $\|L\|_{L^2(P)}=0$, then $L=0$ $P$-a.e., and the Lipschitz bound gives $f_\theta=f_{\theta_*}$ $P$-a.e. for every $\theta\in\Theta$. In this case $\mathcal F$ is covered by the single bracket $[f_{\theta_*},f_{\theta_*}]$ of $L^2(P)$-width $0$. Assume now that $\|L\|_{L^2(P)}>0$, and define \begin{align*} \delta_\varepsilon:=\frac{\varepsilon}{2\|L\|_{L^2(P)}}. \end{align*} Let $\theta_1,\dots,\theta_m\in\Theta$ be a finite $\delta_\varepsilon$-net for $\Theta$ in the Euclidean norm; such a net exists because $\Theta$ is compact. For each $1\le j\le m$, define [measurable functions](/page/Measurable%20Functions) \begin{align*} \ell_j:E\to\mathbb R \end{align*} and \begin{align*} u_j:E\to\mathbb R \end{align*} by \begin{align*} \ell_j(x):=f_{\theta_j}(x)-\delta_\varepsilon L(x) \end{align*} and \begin{align*} u_j(x):=f_{\theta_j}(x)+\delta_\varepsilon L(x). \end{align*} Since $f_{\theta_j}\in L^2(E,\mathcal E,P)$ and $L\in L^2(E,\mathcal E,P)$, both $\ell_j$ and $u_j$ belong to $L^2(E,\mathcal E,P)$. For every $\theta\in\Theta$, choose $j\in\{1,\dots,m\}$ such that $|\theta-\theta_j|\le\delta_\varepsilon$. For every $x\in E\setminus N$, \begin{align*} |f_\theta(x)-f_{\theta_j}(x)|\le \delta_\varepsilon L(x). \end{align*} Hence \begin{align*} \ell_j(x)\le f_\theta(x)\le u_j(x) \end{align*} for $P$-a.e. $x\in E$. The $L^2(P)$-width of this bracket is \begin{align*} \|u_j-\ell_j\|_{L^2(P)}=\|2\delta_\varepsilon L\|_{L^2(P)}=\varepsilon. \end{align*} Thus every $\delta_\varepsilon$-net of $\Theta$ yields an $L^2(P)$-bracketing cover of $\mathcal F$ with bracket width at most $\varepsilon$.[/step]

[guided]The point of this step is to convert metric control in parameter space into metric control in function space. Fix $\varepsilon>0$. If $\|L\|_{L^2(P)}=0$, then $L=0$ $P$-a.e. Since \begin{align*} |f_\theta(x)-f_{\theta_*}(x)|\le L(x)|\theta-\theta_*| \end{align*} for every $x\in E\setminus N$, it follows that $f_\theta=f_{\theta_*}$ $P$-a.e. for every $\theta\in\Theta$. Therefore the single bracket $[f_{\theta_*},f_{\theta_*}]$ covers the whole class in $L^2(P)$ and has width $0$. Now suppose $\|L\|_{L^2(P)}>0$. We want the vertical size of each bracket to be at most $\varepsilon$ in $L^2(P)$. Since a bracket centered at $f_{\theta_j}$ with half-width $\delta L$ has total width $2\delta L$, we choose \begin{align*} \delta_\varepsilon:=\frac{\varepsilon}{2\|L\|_{L^2(P)}}. \end{align*} Compactness of $\Theta$ gives a finite set $\{\theta_1,\dots,\theta_m\}\subset\Theta$ such that every $\theta\in\Theta$ satisfies $|\theta-\theta_j|\le\delta_\varepsilon$ for at least one $j$. For each center $\theta_j$, define \begin{align*} \ell_j:E\to\mathbb R \end{align*} and \begin{align*} u_j:E\to\mathbb R \end{align*} by \begin{align*} \ell_j(x):=f_{\theta_j}(x)-\delta_\varepsilon L(x) \end{align*} and \begin{align*} u_j(x):=f_{\theta_j}(x)+\delta_\varepsilon L(x). \end{align*} These are measurable functions because $f_{\theta_j}$ and $L$ are measurable, and they are in $L^2(P)$ because both summands are in $L^2(P)$. Take any $\theta\in\Theta$ and choose $j$ with $|\theta-\theta_j|\le\delta_\varepsilon$. The Lipschitz condition gives, for every $x\in E\setminus N$, \begin{align*} |f_\theta(x)-f_{\theta_j}(x)|\le L(x)|\theta-\theta_j|\le \delta_\varepsilon L(x). \end{align*} This inequality is equivalent to \begin{align*} f_{\theta_j}(x)-\delta_\varepsilon L(x)\le f_\theta(x)\le f_{\theta_j}(x)+\delta_\varepsilon L(x), \end{align*} so $\ell_j\le f_\theta\le u_j$ outside the $P$-null set $N$. Finally, \begin{align*} \|u_j-\ell_j\|_{L^2(P)}=\|2\delta_\varepsilon L\|_{L^2(P)}=2\delta_\varepsilon\|L\|_{L^2(P)}=\varepsilon. \end{align*} Thus Euclidean nets of the parameter set produce $L^2(P)$-brackets of exactly the desired size.[/guided]

[step:Bound the bracketing entropy by a polynomial covering estimate] For $\delta>0$, let $N(\Theta,|\cdot|,\delta)$ denote the least cardinality of a finite $\delta$-net of $\Theta$ in the Euclidean norm. For $\varepsilon>0$, let $N_{[]}(\varepsilon,\mathcal F,L^2(P))$ denote the least cardinality of a collection of brackets $[\ell,u]$ with $\ell,u\in L^2(E,\mathcal E,P)$, $\ell\le u$ $P$-a.e., and $\|u-\ell\|_{L^2(P)}\le\varepsilon$, whose union contains every $f\in\mathcal F$ up to $P$-a.e. equality. Let $\mathcal L^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $(0,1)$. The preceding step proves that, for every $\varepsilon>0$ with $\|L\|_{L^2(P)}>0$, \begin{align*} N_{[]}(\varepsilon,\mathcal F,L^2(P))\le N\left(\Theta,|\cdot|,\frac{\varepsilon}{2\|L\|_{L^2(P)}}\right). \end{align*} Since $\Theta$ is compact in $\mathbb R^d$, it is bounded. Choose $R>0$ such that $\Theta\subset \overline{B}(0,R)$. A standard grid covering of $\overline{B}(0,R)$ in $\mathbb R^d$ gives a constant $C_\Theta\ge 1$, depending only on $d$ and $R$, such that for every $0<\delta\le 1$, \begin{align*} N(\Theta,|\cdot|,\delta)\le C_\Theta\delta^{-d}. \end{align*} Consequently, for $0<\varepsilon\le 2\|L\|_{L^2(P)}$, \begin{align*} \log N_{[]}(\varepsilon,\mathcal F,L^2(P))\le \log C_\Theta+d\log\left(\frac{2\|L\|_{L^2(P)}}{\varepsilon}\right). \end{align*} The function \begin{align*} \varepsilon\mapsto \sqrt{\log C_\Theta+d\log\left(\frac{2\|L\|_{L^2(P)}}{\varepsilon}\right)} \end{align*} is integrable on every interval $(0,a]$ with $a>0$. If $2\|L\|_{L^2(P)}<1$, then on the remaining interval $(2\|L\|_{L^2(P)},1)$ the bracketing number is bounded above by its value at $2\|L\|_{L^2(P)}$, because bracketing numbers are nonincreasing in the bracket width. Hence the bracketing entropy integral required by the bracketing Donsker theorem is finite: \begin{align*} \int_0^1 \sqrt{\log N_{[]}(\varepsilon,\mathcal F,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty. \end{align*} If $\|L\|_{L^2(P)}=0$, the same conclusion holds because the bracketing number is $1$ for every $\varepsilon>0$. [/step]

[step:Apply the bracketing Donsker theorem] The class $\mathcal F$ is pointwise measurable by hypothesis. We have proved that the measurable function $F:E\to[0,\infty)$ defined by $F(x)=|f_{\theta_*}(x)|+D_\Theta L(x)$ is an envelope for $\mathcal F$ and satisfies $F\in L^2(E,\mathcal E,P)$. We have also proved that \begin{align*} \int_0^1 \sqrt{\log N_{[]}(\varepsilon,\mathcal F,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty. \end{align*} Therefore the hypotheses of the bracketing Donsker criterion are satisfied. By [citetheorem:9836], $\mathcal F$ is $P$-Donsker. This is the desired conclusion. [/step]