In A Course in Real Analysis, J.McDonald, N.Weiss (2004), p.233:
Note:
Let $(\Gamma, \mathcal{S})$ and $(\Lambda, \mathcal{T})$ be measurable spaces. A subset of $\Gamma \times \Lambda$ of the form $S \times T$, where $S \in \mathcal{S}$ and $T \in \mathcal{T}$ is called a measurable rectangle. The collection of all measurable rectangles is denoted by $\mathcal{U}$.
Now there is this Lemma:
$\textbf{Lemma 4.6}$ : Suppose that $\{C_k\}^{n}_{k=1}$ is a finite sequence of pairwise disjoint members of $\mathcal{U}$ whose union is in $\mathcal{U}$. Then $$\iota\left(\bigcup^{n}_{k=1}C_k\right) = \sum_{k=1}^{n} \iota(C_k),$$ where $\iota(C_k)$ is the measure of set $C_k$.
The proof is as following:
Set $C = \bigcup\limits_{k=1}^n C_k$. Then by assumption, we can write $C = S \times T$ and $C_k = S_k \times T_k$ (the Cartesian product), for $1 \leq k\leq n$, where $S, S_k \in \mathcal{S}$ and $T, T_k \in \mathcal{T}$.
Question:
Could anyone give me an intuitive example of $C, C_k, S_k, T_k, S, T$?