2 \sum_{{\alpha },{\beta }=1}^2 e_{2n}(J_{{\alpha }{\beta }}S_{N,g_{\alpha }}R_{\beta },B_{L_\infty^N(\C)}).$$ Moreover, by Proposition \[pro:1\], $$\label{QV5}
e_{2n}(J_{{\alpha }{\beta }}S_{N,g_{\alpha }}R_{\beta },B_{L_\infty^N(\C)})\le e_{n}(S_{N,g_{\alpha }},B_{L_\infty^N({{\mathbb R}})})\quad ({\alpha },{\beta }=1,2).$$ Using the result for the real case and (\[QV2\]), (\[QV4\]), and (\[QV5\]), we get $$e_{2\nu n}(S_{N,g},B_{L_\infty^N(\C)})\le cn^{-1}\|g\|_{L_1^N(\C)},$$ and a scaling of the index concludes the proof.

Now we pass to the case of weighted integration. Let $Q\subseteq {{\mathbb R}}^d$ be a closed, bounded set of positive Lebesgue measure. $L_1(Q)$ denotes the space of Lebesgue integrable functions on $Q$ with values in ${{\mathbb K}}$, equipped with the norm $$\|f\|_{L_1(Q)}=\int_Q |f(x)|\,dx,$$ and $L_\infty(Q)$ the space of all ${{\mathbb K}}$-valued measurable and essentially bounded with respect to the Lebesgue measure functions on $Q$, endowed with the norm $$\|f\|_{\infty}=\mbox{ess\,sup}_{x\in Q} |f(x)|.$$ Let $g\in L_1(Q)$. Define $I_{Q,g}:L_\infty(Q)\to {{\mathbb K}}$, the integration operator with weight $g$, by $$I_{Q,g}f=\int_Q g(x)f(x)\,dx.$$ $C(Q)$ denotes the space of continuous functions on $Q$, equipped with the supremum norm. A set $\mathcal{E}$ of continuous functions on $Q$ is called uniformly equicontinuous, if for each ${\varepsilon }>0$ there is a $\delta>0$ such that for $x,y\in Q$, $|x-y|\le\delta$ implies $|f(x)-f(y)|\le {\varepsilon }$ for all $f\in F$. By the Arzelà-Ascoli theorem, bounded, uniformly equicontinuous sets coincide with precompact subsets of 