the so-called *Hardy–Littlewood–Pólya relation*, defined by $$\begin{aligned}
\int _{0}^{t}f^{*}(s){\mathrm{d}}s\leq \int _{0}^{t}g^{*}(s){\mathrm{d}}s, \quad t\in (0,\infty ),\end{aligned}$$ and sometimes denoted by $f\prec g$ in the literature, then $\|f\|_{X}\leq \|g\|_{X}$ provided that the underlying measure space is resonant. We note that throughout this paper we work solely on nonatomic measure spaces, which are resonant by [@BS Chapter 2, Theorem 2.7].

For every r.i. space $X$ over the measure space $(R,\mu )$ there exists a unique rearrangement-invariant space $X(0,\mu (R))$ over the interval $(0,\mu (R))$ endowed with the one-dimensional Lebesgue measure such that $\|f\|_{X}=\|f^{*}\|_{X(0,
\mu(R))}$. This space is called the *representation space* of $X$. This follows from the Luxemburg representation theorem [@BS Chapter 2, Theorem 4.10]. Throughout this paper, the representation space of an r.i. space $X$ will be denoted by $X(0,\mu (R))$. It will be useful to notice that when $R=(0,\infty )$ and $\mu $ is the Lebesgue measure, then every $X$ over $(R,\mu )$ coincides with its representation space.

If $\varrho $ is an r.i. norm and $X=X({\varrho })$ is the r.i. space determined by $\varrho $, we define its *fundamental function*, $\varphi _{X}$, for every $t\in [0,\mu (R))$ by $\varphi _{X}(t)=\varrho (\chi _{E})$, where $E\subset R$ is such that $\mu (E)=t$. 