[proofplan] We apply the bounded empirical-process maximal inequality to the localized class $\mathcal F(r)$. The localization hypothesis gives the variance radius $r$, while the original envelope $F$ remains an envelope for the subclass and is bounded by $M$. Since passing to a subclass cannot increase uniform covering numbers, the entropy integral of $\mathcal F(r)$ is bounded by the entropy integral of $\mathcal F$. Finally, the maximal inequality controls the empirical process $G_n(f)=\sqrt n(P_n-P)f$, so dividing by $\sqrt n$ gives the stated bound. [/proofplan] [step:Introduce the empirical process and localize the class] Define the empirical process \begin{align*} G_n:\mathcal F\to\mathbb R \end{align*} by \begin{align*} G_n(f):=\sqrt n\,(P_n-P)f \end{align*} for $f\in\mathcal F$. The localized class $\mathcal F(r)$ is pointwise measurable by hypothesis, and the measurability of the displayed supremum is assumed in the statement. For every $f\in\mathcal F(r)$, the definition of $\mathcal F(r)$ gives \begin{align*} P f^2\le r^2. \end{align*} Moreover, since $|f|\le F\le M$ for every $f\in\mathcal F$, the same function $F:S\to[0,M]$ is an envelope for $\mathcal F(r)$. [/step] [step:Compare the entropy integral of the localized class with that of the original class] For a finitely supported probability measure $Q$ on $(S,\mathcal S)$, a class $\mathcal G$ of [measurable functions](/page/Measurable%20Functions) on $S$, and a radius $a>0$, let $N(a,\mathcal G,L^2(Q))$ denote the least cardinality of an $L^2(Q)$-ball cover of $\mathcal G$ by balls of radius $a$. For every finitely supported probability measure $Q$ on $(S,\mathcal S)$ and every $\varepsilon>0$, every $L^2(Q)$-cover of $\mathcal F$ by balls of radius $\varepsilon\|F\|_{L^2(Q)}$ also covers the subclass $\mathcal F(r)$. Hence \begin{align*} N\left(\varepsilon\|F\|_{L^2(Q)},\mathcal F(r),L^2(Q)\right)\le N\left(\varepsilon\|F\|_{L^2(Q)},\mathcal F,L^2(Q)\right). \end{align*} Taking logarithms, suprema over finitely supported probability measures $Q$, square roots, and then integrating over the entropy parameter in the definition of the uniform entropy integral gives \begin{align*} J\left(\frac{r}{M},\mathcal F(r),F\right)\le J\left(\frac{r}{M},\mathcal F,F\right). \end{align*} [guided] The only point in this step is that localization reduces the index set but does not change the envelope. Fix a finitely supported probability measure $Q$ on $(S,\mathcal S)$ and a number $\varepsilon>0$. The $L^2(Q)$-covering number \begin{align*} N\left(\varepsilon\|F\|_{L^2(Q)},\mathcal F,L^2(Q)\right) \end{align*} is the smallest number of $L^2(Q)$-balls of radius $\varepsilon\|F\|_{L^2(Q)}$ needed to cover $\mathcal F$. Since $\mathcal F(r)\subseteq\mathcal F$, any family of balls that covers $\mathcal F$ also covers $\mathcal F(r)$. Therefore \begin{align*} N\left(\varepsilon\|F\|_{L^2(Q)},\mathcal F(r),L^2(Q)\right)\le N\left(\varepsilon\|F\|_{L^2(Q)},\mathcal F,L^2(Q)\right). \end{align*} The logarithm is increasing on $(0,\infty)$, and the square-root function is increasing on $[0,\infty)$. Hence the integrand defining the uniform entropy integral for $\mathcal F(r)$ is bounded pointwise by the corresponding integrand for $\mathcal F$. After taking the supremum over all finitely supported probability measures $Q$ and integrating over the interval from $0$ to $r/M$, we obtain \begin{align*} J\left(\frac{r}{M},\mathcal F(r),F\right)\le J\left(\frac{r}{M},\mathcal F,F\right). \end{align*} This is the reason the final estimate can be written using the entropy of the original class rather than a new entropy quantity for the localized subclass. [/guided] [/step] [step:Apply the bounded empirical process maximal inequality] We apply the [citetheorem:9842] to the class $\mathcal F(r)$ with envelope $F$, envelope bound $M$, and variance radius $\sigma=r$. Its hypotheses are satisfied: the random variables $X_1,\dots,X_n$ are independent identically distributed with common distribution $P$; the functions in $\mathcal F(r)$ are measurable; the class $\mathcal F(r)$ is pointwise measurable by hypothesis; the envelope satisfies $F\le M$; the range condition $r\le M$ holds because $r\in(0,M]$; every $f\in\mathcal F(r)$ satisfies $P f^2\le r^2$; and the entropy condition for $\mathcal F(r)$ is finite because the previous step gives $J(r/M,\mathcal F(r),F)\le J(r/M,\mathcal F,F)<\infty$. Therefore there exists a universal constant $C_0>0$ such that \begin{align*} \mathbb E\left[\sup_{f\in\mathcal F(r)}|G_n(f)|\right]\le C_0\left\{M J\left(\frac{r}{M},\mathcal F(r),F\right)+\frac{M^2}{\sqrt n\,r^2}J\left(\frac{r}{M},\mathcal F(r),F\right)^2\right\}. \end{align*} Using the entropy comparison from the previous step yields \begin{align*} \mathbb E\left[\sup_{f\in\mathcal F(r)}|G_n(f)|\right]\le C_0\left\{M J\left(\frac{r}{M},\mathcal F,F\right)+\frac{M^2}{\sqrt n\,r^2}J\left(\frac{r}{M},\mathcal F,F\right)^2\right\}. \end{align*} [/step] [step:Divide by the empirical-process normalization] For every $f\in\mathcal F(r)$, the definition of $G_n$ gives \begin{align*} |(P_n-P)f|=\frac{1}{\sqrt n}|G_n(f)|. \end{align*} Taking suprema over $f\in\mathcal F(r)$ and then expectations gives \begin{align*} \mathbb E\left[\sup_{f\in\mathcal F(r)}|(P_n-P)f|\right]=\frac{1}{\sqrt n}\mathbb E\left[\sup_{f\in\mathcal F(r)}|G_n(f)|\right]. \end{align*} Combining this identity with the bound from the preceding step gives \begin{align*} \mathbb E\left[\sup_{f\in\mathcal F(r)}|(P_n-P)f|\right]\le C_0\left\{\frac{M}{\sqrt n}J\left(\frac{r}{M},\mathcal F,F\right)+\frac{M^2}{nr^2}J\left(\frac{r}{M},\mathcal F,F\right)^2\right\}. \end{align*} Setting $C:=C_0$ proves the desired estimate with a universal constant $C>0$. [/step]