I would like to notate something to the effect of the following—
$$\left\{ x \ \mathrm{cm}\,\middle|\,0\lt x\le5\right\}$$
—but as an interval.
Here’s what’s in my mind:
$$\begin{array}{cl} \text{(1)} & (0,5] \ \mathrm{cm} \\ \text{(2)} & (0 \ \mathrm{cm} ,5 \ \mathrm{cm}] \\ \text{(3)} & \text{doesn’t really work} \\ \end{array}$$
$\text{(1)}$ seems less than rigorous, because it’s multiplying an interval by an entity, and that just seems weird. $\text{(2)}$ is clear, but it doesn’t seem sound, because the real line doesn’t include units. Is $\text{(3)}$ my only option?
And as an extra twist, consider
$$\left\{ x \ \mathrm{cm}\,\middle|\,0\lt x\lt\infty\right\}$$
(1) seems fine, and (2) is okay too. They indicate the interval and the units of measurement used over it.
$$(0;5]\mathrm{cm} ~{=~ (0\mathrm{cm};5\mathrm{cm}] \\=~ \{s\mid 0\mathrm{cm}\lt s\leqslant 5\mathrm{cm}\} \\=~ \{x\cdotp\mathrm{cm}\mid 0<x\leqslant 5\}}$$