[proofplan] We prove the countable-class version, which makes all empirical suprema measurable and removes the conditional-expectation null-set obstruction present for arbitrary uncountable classes. An independent ghost sample replaces the deterministic mean $P f$ by the [conditional expectation](/page/Conditional%20Expectation) of a second empirical mean. Exchangeability of each pair $(X_i,X_i')$ inserts Rademacher signs into the ghost difference, and the elementary inequality for a supremum of sums splits the signed ghost process into two Rademacher processes. The two split terms have the same distribution as the original Rademacher supremum, giving the factor $2$. [/proofplan]

[step:Pass to a product extension carrying an independent ghost sample] Let $\mathcal S^{\otimes n}$ denote the product $\sigma$-algebra on $S^n$, and let $P^{\otimes n}$ denote the $n$-fold product probability measure on $(S^n,\mathcal S^{\otimes n})$. Replace the [probability space](/page/Probability%20Space), if necessary, by the product extension \begin{align*} (\widetilde\Omega,\widetilde\mathcal A,\widetilde\mathbb P) := (\Omega\times S^n,\mathcal A\otimes\mathcal S^{\otimes n},\mathbb P\otimes P^{\otimes n}). \end{align*} On this extension, the original random variables are identified with their pullbacks under the first coordinate projection. This identification preserves their joint distribution, hence preserves every outer expectation involving only $X_1,\dots,X_n$ and $\varepsilon_1,\dots,\varepsilon_n$. For $i\in\{1,\dots,n\}$, define the ghost [random variable](/page/Random%20Variable) \begin{align*} X_i':(\widetilde\Omega,\widetilde\mathcal A)&\to(S,\mathcal S) \end{align*} by taking the $i$-th coordinate of the $S^n$ factor. Then $X_1',\dots,X_n'$ are independent, have common distribution $P$, and are independent of $X_1,\dots,X_n,\varepsilon_1,\dots,\varepsilon_n$. For $f\in\mathcal F$, define the ghost empirical mean by \begin{align*} P_n'f:=\frac{1}{n}\sum_{i=1}^{n}f(X_i'). \end{align*} Since $f$ is $P$-integrable, each $f(X_i)$ and $f(X_i')$ is an integrable real-valued random variable. Because $\mathcal F$ is countable, all suprema over $\mathcal F$ appearing below are measurable extended-real random variables. [/step]

[step:Bound the original one-sided process by the ghost difference]Let \begin{align*} \mathcal G:=\sigma(X_1,\dots,X_n) \end{align*} denote the $\sigma$-algebra generated by the original sample. For each $f\in\mathcal F$, independence of $X_1',\dots,X_n'$ from $\mathcal G$ and the identity $X_i'\sim P$ imply \begin{align*} \mathbb E[P_n'f\mid\mathcal G]=P f. \end{align*} Define the measurable extended-real random variables \begin{align*} Z:=\sup_{f\in\mathcal F}(P f-P_n f) \end{align*} and \begin{align*} Y:=\sup_{f\in\mathcal F}(P_n'f-P_n f). \end{align*} If $\mathbb E^*[Y]=+\infty$, then the desired comparison with $Y$ is immediate. Otherwise, let $U$ be an arbitrary integrable measurable random variable such that $U\ge Y$ almost surely. For every fixed $f\in\mathcal F$, monotonicity of conditional expectation gives \begin{align*} \mathbb E[U\mid\mathcal G]\ge \mathbb E[P_n'f-P_n f\mid\mathcal G]=P f-P_n f \end{align*} almost surely. Since $\mathcal F$ is countable, the exceptional null sets may be united into one null set, so the preceding inequality holds simultaneously for every $f\in\mathcal F$ outside that null set. Hence \begin{align*} \mathbb E[U\mid\mathcal G]\ge Z \end{align*} almost surely. The definition of outer expectation therefore yields \begin{align*} \mathbb E^*[Z]\le \mathbb E[\mathbb E[U\mid\mathcal G]]=\mathbb E[U]. \end{align*} Taking the infimum over all integrable measurable majorants $U$ of $Y$ gives \begin{align*} \mathbb E^*\left[\sup_{f\in\mathcal F}(P f-P_n f)\right] \le \mathbb E^*\left[\sup_{f\in\mathcal F}(P_n'f-P_n f)\right]. \end{align*}[/step]

[guided]The role of the ghost sample is to express $P f$ as a conditional average. Let \begin{align*} \mathcal G:=\sigma(X_1,\dots,X_n). \end{align*} For a fixed $f\in\mathcal F$, the random variables $f(X_i')$ are integrable because $X_i'\sim P$ and \begin{align*} \int_S |f(x)|\,dP(x)<\infty. \end{align*} Since the ghost sample is independent of $\mathcal G$, conditional expectation gives \begin{align*} \mathbb E[f(X_i')\mid\mathcal G]=\mathbb E[f(X_i')]=\int_S f(x)\,dP(x)=P f. \end{align*} Averaging this identity over $i\in\{1,\dots,n\}$ gives \begin{align*} \mathbb E[P_n'f\mid\mathcal G]=P f. \end{align*} Consequently, \begin{align*} \mathbb E[P_n'f-P_n f\mid\mathcal G]=P f-P_n f, \end{align*} because $P_n f$ is $\mathcal G$-measurable and integrable. Now define \begin{align*} Z:=\sup_{f\in\mathcal F}(P f-P_n f) \end{align*} and \begin{align*} Y:=\sup_{f\in\mathcal F}(P_n'f-P_n f). \end{align*} The countability of $\mathcal F$ is essential here: it makes $Y$ and $Z$ measurable as countable suprema, and it lets us combine fixed-$f$ almost-sure inequalities into one simultaneous almost-sure inequality. If $\mathbb E^*[Y]=+\infty$, then \begin{align*} \mathbb E^*[Z]\le \mathbb E^*[Y] \end{align*} holds with no further work. Otherwise, choose an arbitrary integrable measurable random variable $U$ with $U\ge Y$ almost surely. Then, for every fixed $f\in\mathcal F$, \begin{align*} U\ge P_n'f-P_n f \end{align*} almost surely. Monotonicity of conditional expectation with respect to $\mathcal G$ gives \begin{align*} \mathbb E[U\mid\mathcal G] \ge \mathbb E[P_n'f-P_n f\mid\mathcal G] = P f-P_n f \end{align*} almost surely. For one fixed $f$, this inequality may fail on a null set depending on $f$. Since $\mathcal F$ is countable, the union of all these exceptional null sets is still null. Outside that one null set, the inequality holds for every $f\in\mathcal F$, and therefore \begin{align*} \mathbb E[U\mid\mathcal G]\ge \sup_{f\in\mathcal F}(P f-P_n f)=Z. \end{align*} Thus $\mathbb E[U\mid\mathcal G]$ is an integrable measurable majorant of $Z$. By the definition of outer expectation, \begin{align*} \mathbb E^*[Z]\le \mathbb E[\mathbb E[U\mid\mathcal G]]=\mathbb E[U]. \end{align*} Finally, because $U$ was an arbitrary integrable measurable majorant of $Y$, taking the infimum over all such $U$ yields \begin{align*} \mathbb E^*\left[\sup_{f\in\mathcal F}(P f-P_n f)\right] \le \mathbb E^*\left[\sup_{f\in\mathcal F}(P_n'f-P_n f)\right]. \end{align*}[/guided]

[step:Insert Rademacher signs using exchangeability of each original and ghost pair] For each $i\in\{1,\dots,n\}$, the pair $(X_i',X_i)$ has the same distribution as $(X_i,X_i')$. The pairs are independent, and the signs $\varepsilon_1,\dots,\varepsilon_n$ are independent of all original and ghost variables. Therefore the random array \begin{align*} (X_1',X_1,\dots,X_n',X_n) \end{align*} has the same distribution as the random array obtained by swapping the two entries of the $i$-th pair exactly when $\varepsilon_i=-1$. Define \begin{align*} T:=\sup_{f\in\mathcal F}(P_n'f-P_n f) \end{align*} and \begin{align*} T_\varepsilon:=\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i\{f(X_i')-f(X_i)\}. \end{align*} The preceding exchangeability shows that $T$ and $T_\varepsilon$ have the same distribution. Since both are measurable extended-real random variables, their measurable majorants have the same laws under composition with the corresponding sample-array maps, and hence \begin{align*} \mathbb E^*[T]=\mathbb E^*[T_\varepsilon]. \end{align*} [/step]

[step:Split the signed ghost process into two Rademacher suprema] For each $f\in\mathcal F$, define the real-valued random variables \begin{align*} A_f:=\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i') \end{align*} and \begin{align*} B_f:=\frac{1}{n}\sum_{i=1}^{n}(-\varepsilon_i)f(X_i). \end{align*} Then \begin{align*} \frac{1}{n}\sum_{i=1}^{n}\varepsilon_i\{f(X_i')-f(X_i)\}=A_f+B_f. \end{align*} For every outcome, the order property of the supremum gives \begin{align*} \sup_{f\in\mathcal F}(A_f+B_f)\le \sup_{f\in\mathcal F}A_f+\sup_{f\in\mathcal F}B_f. \end{align*} If either of the two outer expectations on the right below is $+\infty$, the desired inequality is immediate. Otherwise, take arbitrary integrable measurable majorants $U$ and $V$ of $\sup_{f\in\mathcal F}A_f$ and $\sup_{f\in\mathcal F}B_f$, respectively. Then $U+V$ is an integrable measurable majorant of $T_\varepsilon$. Taking expectations and then infima over $U$ and $V$ gives \begin{align*} \mathbb E^*[T_\varepsilon] \le \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i')\right] + \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}(-\varepsilon_i)f(X_i)\right]. \end{align*} [/step]

[step:Identify both split terms with the original Rademacher supremum] The sign vector $(-\varepsilon_1,\dots,-\varepsilon_n)$ has the same distribution as $(\varepsilon_1,\dots,\varepsilon_n)$ and is independent of $X_1,\dots,X_n$. Hence \begin{align*} \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}(-\varepsilon_i)f(X_i)\right] = \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i)\right]. \end{align*} Also, $(X_1',\dots,X_n')$ has the same distribution as $(X_1,\dots,X_n)$ and is independent of $(\varepsilon_1,\dots,\varepsilon_n)$. Therefore \begin{align*} \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i')\right] = \mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i)\right]. \end{align*} Combining the ghost comparison, the exchangeability identity, the splitting estimate, and these two distributional identities yields \begin{align*} \mathbb E^*\left[\sup_{f\in\mathcal F}(P f-P_n f)\right] \le 2\,\mathbb E^*\left[\sup_{f\in\mathcal F}\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i f(X_i)\right]. \end{align*} This is the asserted one-sided symmetrisation inequality. [/step]