The discussion in the comments has triggered my curiosity: I've done a little bit of research and I almost immediately stumbled upon the [Wikipedia entry on the Peano-Jordan measure](https://en.wikipedia.org/wiki/Peano–Jordan_measure) and there I found exactly the terms "*simple sets*" and "*polyrectangles*" used as synonyms for the unions of finite families of $n$-rectangles, i.e. for the set $$ S=\bigcup_{i=1}^q C_i\quad q\in\Bbb N_{>0} $$ where $n$ is the dimension of the Euclidean $\Bbb R^n$ considered and $C_i$ are $n$-rectangles defined as $C_i=[a_{1i}, b_{1i})\times\cdots\times[a_{ni}, b_{ni})$. However, I was not able to find where these sets are called in this way within the references stated in the relevant section of the entry. Thus I started searching in my personal library and then I found the locution "*plurirectangles*" (*plurirettangoli* in Italian) in the classical monograph \[1], chapter II, §2.7, p. 119, as shown in the picture below: [![enter image description here][1]][1] This makes me sufficiently confident in proposing the adoption of this, somewhat traditional, terminology. **Reference** \[1] Federico Cafiero (1959), *Misura e integrazione* [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, [MR0215954](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0215954), [Zbl 0171.01503](https://www.zbmath.org/?q=an%3A0171.01503). [1]: https://i.sstatic.net/Wi7VtrVw.jpg