Oppositely monotone functions of independent coordinates have nonpositive covariance

In dimension one, let $\mu$ be the law of $U_1$. By [F7], the expectations defining the covariance are the corresponding $\mu$-integrals. Expanding those integrals using [F8] and applying [F3] gives $2\mathrm{Cov}(G(U_1),H(U_1))=\int \text{\hspace{0.17em}⁣}\int (G(x)-G(y))(H(x)-H(y))\mu (dx)\mu (dy)$. The integrand is nonpositive for every $x,y$ because the two functions have opposite monotonicity, so the covariance is nonpositive. The two differences are square-integrable on $({\mathbb{R}}^2,\mu \times \mu )$, so [F4] makes their product integrable and justifies the double integral.

Assume the claim holds for $d-1$ and first suppose $G,H$ are bounded. By [F2] the joint law is $\mu \times \nu$, where $\mu$ is the first $d-1$ coordinate laws and $\nu$ is the law of $U_d$. The change-of-variables formula [F7] identifies expectations under the joint law with integrals against $\mu \times \nu$. Define $G_0(x)=\int G(x,z)\nu (dz)$ and $H_0(x)=\int H(x,z)\nu (dz)$. Fubini [F3] makes these measurable and bounded; they retain their respective coordinate monotonicities. For each fixed $x$, step 1.1 applied to the last coordinate gives ${\mathrm{Cov}}_{\nu }(G(x,U_d),H(x,U_d))\le 0$, and the induction hypothesis gives ${\mathrm{Cov}}_{\mu }(G_0,H_0)\le 0$.

Expanding both sides by the covariance formula [F5], using expectation linearity [F8], the joint law identity [F7], and Fubini [F3] yields $\mathrm{Cov}(G(U),H(U))=\int {\mathrm{Cov}}_{\nu }(G(x,U_d),H(x,U_d))\mu (dx)+{\mathrm{Cov}}_{\mu }(G_0,H_0)\le 0$. Thus the bounded case holds in dimension $d$, completing the induction.

For general square-integrable $G,H$, clamp each function to $[-m,m]$. The clamps remain coordinatewise monotone and bounded, with $\mathrm{\mid }{G}_m\mathrm{\mid }\le \mathrm{\mid }G\mathrm{\mid }$, $\mathrm{\mid }{H}_m\mathrm{\mid }\le \mathrm{\mid }H\mathrm{\mid }$, $\mathrm{\mid }{G}_m-G{\mathrm{\mid }}^2\le G^2$, and $\mathrm{\mid }{H}_m-H{\mathrm{\mid }}^2\le H^2$. Applying [F6] to the squared differences gives $L^2$ convergence, while $\mathrm{\parallel }{G}_m{\mathrm{\parallel }}_2\le \mathrm{\parallel }G{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }{H}_m{\mathrm{\parallel }}_2\le \mathrm{\parallel }H{\mathrm{\parallel }}_2$. Cauchy–Schwarz [F4] shows that $E\mathrm{\mid }{G}_m-G\mathrm{\mid }$ and $E\mathrm{\mid }{H}_m-H\mathrm{\mid }$ tend to zero, and that $E\mathrm{\mid }{G}_m{H}_m-GH\mathrm{\mid }\le \mathrm{\parallel }{G}_m-G{\mathrm{\parallel }}_2\mathrm{\parallel }{H}_m{\mathrm{\parallel }}_2+\mathrm{\parallel }G{\mathrm{\parallel }}_2\mathrm{\parallel }{H}_m-H{\mathrm{\parallel }}_2$ tends to zero. Thus expectation linearity [F8] and the covariance identity [F5] imply $\mathrm{Cov}(G_m(U),H_m(U))\to \mathrm{Cov}(G(U),H(U))$. Each bounded covariance is nonpositive by step 3.1, so the limit is nonpositive.