1\right \}  .\end{aligned}$$ In other words, for such $f$, the supremum can be reduced to nonincreasing functions only without any loss of information. This fact has deep consequences and will be used in the proofs below.

For each $a\in (0,\infty )$, let $D_{a}$ denote the *dilation operator* defined on every nonnegative measurable function $f$ on $(0,\infty )$ by $$\begin{aligned}
(D_{a}f)(t)=f(at),\quad t\in (0,\infty ).\end{aligned}$$ The operator $D_{a}$ is bounded on every rearrangement-invariant space over $(0,\infty )$ (hence in particular on the representation space of any r.i. space over an arbitrary adequate measure space). More precisely, if $X$ is any given r.i. space over $(0,\infty )$ with respect to the one-dimensional Lebesgue measure, then we have $$\begin{aligned}
\|D_{a}f\|_{X}\leq C\|f\|_{X}, \quad f\in X,\end{aligned}$$ with $C\le \max\{1,\frac{1}{a}\}$. For more details, see [@BS Chapter 3, Proposition 5.11].

Among basic examples of function norms are those associated with the standard Lebesgue spaces $L^{p}$. For $p\in (0,\infty ]$, we define the functional $\varrho _{p}$ by $$\begin{aligned}
\varrho _{p}(f)=\|f\|_{p}=
{\begin{cases}
\left (\int _{R}f^{p}\,{\mathrm{d}}\mu \right )^{\frac{1}{p}}
&\textup{if}\ 0<p<\infty ,
\cr
\operatorname{ess\,\sup }_{R}f
&\textup{if}\ p=\infty
\end{cases}}
$$ for $f \in \mathcal{M}_{+}(R,\mu )$. If $p\in [1,\infty ]$, then $\varrho _{p}$ is an r.i. norm.

If $0< p,q\le \infty $, we define 