Independent-process symmetrization for convex moments

First let ${\mathcal{F}}_0\subset \mathcal{F}$ be finite. For fixed observed values $z_1,\dots ,z_n$, write $D_f:={\sum }_i(f(z_i)-f(Z_i^{\mathrm{\prime }}))$. Then ${\sum }_i(f(z_i)-Pf)={\mathbb{E}}_{Z^{\mathrm{\prime }}}{D}_f$. For each $f$, $\mathrm{\mid }{\mathbb{E}}_{Z^{\mathrm{\prime }}}{D}_f\mathrm{\mid }\le {\mathbb{E}}_{Z^{\mathrm{\prime }}}\mathrm{\mid }{D}_f\mathrm{\mid }\le {\mathbb{E}}_{Z^{\mathrm{\prime }}}{\max }_{g\in {\mathcal{F}}_0}\mathrm{\mid }{D}_g\mathrm{\mid }$; taking the finite maximum and applying monotonicity of $\mathrm{\Phi }$ and Jensen [F3] gives $\mathrm{\Phi }({\max }_f\mathrm{\mid }{\sum }_i(f(z_i)-Pf)\mathrm{\mid })\le {\mathbb{E}}_{Z^{\mathrm{\prime }}}\mathrm{\Phi }({\max }_f\mathrm{\mid }{D}_f\mathrm{\mid })$. By [F8], these finite statistics and their $\mathrm{\Phi }$-compositions are measurable, so Tonelli [F4] applies when integrating over observations and copy. It gives $\mathbb{E}\mathrm{\Phi }({\max }_f\mathrm{\mid }{\sum }_i(f(Z_i)-Pf)\mathrm{\mid })\le {\mathbb{E}}_{Z,Z^{\mathrm{\prime }}}\mathrm{\Phi }({\max }_f\mathrm{\mid }{\sum }_i(f(Z_i)-f(Z_i^{\mathrm{\prime }}))\mathrm{\mid })$.

By [F6], each $Z_i$ and $Z_i^{\mathrm{\prime }}$ has law $P$; by [F2] and the independent-copy hypothesis, all $2n$ coordinates have product law. [F7] makes this product law invariant under swapping the two coordinates in any selected pair. Thus for every fixed sign vector $\sigma$, the signed difference process ${\sum }_i{\sigma }_i(f(Z_i)-f(Z_i^{\mathrm{\prime }}))$ has the same law as the unsigned difference process. Averaging this equality over the finite sign space leaves the right-hand expectation in step 1.1 unchanged when $f(Z_i)-f(Z_i^{\mathrm{\prime }})$ is replaced by ${\sigma }_i(f(Z_i)-f(Z_i^{\mathrm{\prime }}))$.

For every outcome, put $A:={\max }_f\mathrm{\mid }{\sum }_i{\sigma }_{i}f(Z_i)\mathrm{\mid }$ and $B:={\max }_f\mathrm{\mid }{\sum }_i{\sigma }_{i}f(Z_i^{\mathrm{\prime }})\mathrm{\mid }$. By [F8] these statistics and their $\mathrm{\Phi }$-compositions are measurable. The triangle inequality gives ${\max }_f\mathrm{\mid }{\sum }_i{\sigma }_i(f(Z_i)-f(Z_i^{\mathrm{\prime }}))\mathrm{\mid }\le A+B$. Monotonicity and convexity yield $\mathrm{\Phi }(A+B)\le \frac{1}{2}\mathrm{\Phi }(2A)+\frac{1}{2}\mathrm{\Phi }(2B)$; since $A$ and $B$ have the same distribution, the resulting expectation is $\mathbb{E}\mathrm{\Phi }(2A)$. Combining with steps 1.1 and 2.1 proves the desired inequality for finite ${\mathcal{F}}_0$.

Enumerate the countable class and let ${\mathcal{F}}_k$ be its first $k$ members. The finite-class inequality applies to every ${\mathcal{F}}_k$, and its suprema increase to those over $\mathcal{F}$. To justify the limit, fix $c>0$ and $b>c$. For $c<x<b$, monotonicity and convexity give $0\le \mathrm{\Phi }(x)-\mathrm{\Phi }(c)\le \frac{x-c}{b-c}(\mathrm{\Phi }(b)-\mathrm{\Phi }(c))$; for $0<x<c$, writing $c=\frac{b-c}{b-x}x+\frac{c-x}{b-x}b$ gives $0\le \mathrm{\Phi }(c)-\mathrm{\Phi }(x)\le \frac{c-x}{b-c}(\mathrm{\Phi }(b)-\mathrm{\Phi }(c))$. Thus $\mathrm{\Phi }$ is continuous on $(0,\mathrm{\infty })$. At zero, for any $a>0$ and $t\downarrow 0$, monotonicity and convexity give $\mathrm{\Phi }(0)\le \mathrm{\Phi }(ta)\le (1-t)\mathrm{\Phi }(0)+t\mathrm{\Phi }(a)$, so it is right-continuous there too. By [F8], the finite-class compositions are measurable; [F5] now passes both nonnegative expectations to the limit. A countable subclass giving the same suprema proves the stated separable-version extension.