Definition 1

One says that a subset $\mathscr{E}$ of $\mathscr{H}(\mathfrak{F}, K)$ formed of functions not equivalent to 0 modulo $R_\infty$ is a comparison scale when $\mathscr{E}$ is totally ordered by the relation " $f \ll g$ or $f = g$".

In other words, if $f$ and $g$ are functions in $\mathscr{E}$ then one (and only one) of the relations $f \ll g$, $g \ll f$, $f = g$ always holds. It follows that on $\mathscr{E}$ the relation $f \asymp g$ (and a fortiori $|f| \sim a\,|g|$, where $a$ is a number $> 0$) implies $f = g$.

Every subset of a comparison scale is clearly also a comparison scale.