$\varrho $ is associated another functional, $\varrho '$, defined for $g \in \mathcal{M}_{+}(R,
\mu )$ as $$\begin{aligned}
\varrho '(g)=\sup \left \{  \int _{R} fg\,{\mathrm{d}}\mu : f\in \mathcal{M}_{+}(R,\mu ),\ \varrho (f)\leq 1\right \}
.\end{aligned}$$ It turns out that $\varrho '$ is also an r.i. norm, which is called the *associate norm* of $\varrho $. Moreover, for every r.i. norm $\varrho $ and every $f\in
\mathcal{M}_{+}(R,\mu )$, we have (see [@BS Chapter 1, Theorem 2.9]) $$\begin{aligned}
\varrho (f)=\sup \left \{  \int _{R}fg\,{\mathrm{d}}\mu : g\in \mathcal{M}_{+}(R,\mu ),\ \varrho '(g)\leq 1\right \}
.\end{aligned}$$ If $\varrho $ is an r.i. norm, $X=X({\varrho })$ is the r.i. space determined by $\varrho $, and $\varrho '$ is the associate norm of $\varrho $, then the function space $X({\varrho '})$ determined by $\varrho '$ is called the *associate space* of $X$ and is denoted by $X'$. We always have $(X')'=X$, and we shall write $X''$ instead of $(X')'$. Furthermore, the *Hölder inequality* $$\begin{aligned}
\int _{R}fg\,{\mathrm{d}}\mu \leq \|f\|_{X}\|g\|_{X'}\end{aligned}$$ holds for every $f,g\in \mathcal{M}(R,\mu )$.

An important consequence of the Hardy lemma, which plays a crucial role in the theory of rearrangement-invariant spaces, is the *Hardy–Littlewood–Pólya principle* [@BS Chapter 2, Theorem 4.6] which asserts that if two functions $f,g$ satisfy 