$\Gamma_{{\ensuremath{\mathbf b}}^{(2)}}^{(s_2)}$, and use the system $\mathcal{T}^{(s_3)}$ for the final $s_3$ coordinates.

Let $s\in {\ensuremath{\mathbb{N}}}$. By an *$s$-dimensional index set* $\Lambda$ we understand one of the additive semigroups $({\ensuremath{\mathbb{Z}}}^s, +)$, $({\ensuremath{\mathbb{N}}}_0^s, +)$, and $({\ensuremath{\mathbb{N}}}^s, +)$, or finite direct products of these semigroups such that the dimensions of the factors add up to $s$.

Let $\Lambda^*$ denote the index set $\Lambda\setminus \{\mathbf{0}\}$.

Examples of $s$-dimensional index sets are direct products of the form ${\ensuremath{\mathbb{N}}}_0^{s_1}\times {\ensuremath{\mathbb{N}}}_0^{s_2}\times {\ensuremath{\mathbb{Z}}}^{s_3}$, where $s=s_1+s_2+s_3$, with $s_1, s_2, s_3\in {\ensuremath{\mathbb{N}}}_0$, not all $s_i$ equal to 0, as they appear in hybrid function systems.

If $\omega = ({\bf x}_n)_{n\ge 0}$ is a -possibly finite- sequence in $[0,1)^s$ with at least $N$ elements, and if $f:\, [0,1)^s$ $\to$ ${{\ensuremath{\mathbb{C}}}}$, we define $$S_N(f,\omega) = \frac{1}{N} \sum_{n=0}^{N-1} f({\bf x}_n).$$

Let $\Lambda$ be an $s$-dimensional index set. A subclass ${\mathcal{F}}=\{ \xi_{\ensuremath{\mathbf k}}: {\ensuremath{\mathbf k}}\in \Lambda  \}$ of the class of Riemann integrable functions on $[0,1)^s$ is called *a uniform distribution determining (u.d.d.) function system on $[0,1)^s$* if the functions $\xi_{\ensuremath{\mathbf k}}$ are normalized in the sense that $||\xi_{\ensuremath{\mathbf k}}||_\infty = \sup\{|\xi_{\ensuremath{\mathbf k}}({\ensuremath{\mathbf x}})|: {\ensuremath{\mathbf x}}\in [0,1)^s\}\le 1$ for all ${\ensuremath{\mathbf k}}$, and $\int_{[0,1)^s} \xi_{\ensuremath{\mathbf k}}d\lambda_s = 0$ for all ${\ensuremath{\mathbf k}}\in \Lambda^*$, and 