Example 1

For a real function $f \in \mathscr{H}(\mathfrak{F}, \mathbf{R})$ the relation $f \simeq 1$ means that there are two numbers $a > 0,\ b > 0$ such that $a \leq |f(x)| \leq b$ on a set in $\mathfrak{F}$, or that the function $\log |f|$ is bounded on a set in $\mathfrak{F}$: one then says that $f$ is logarithmically bounded on a set in $\mathfrak{F}$.