every $f,g\in \mathcal{M}(R,\mu )$, we have $$\begin{aligned}
\label{E:subadditivity-of-doublestar}
\int _{0}^{t}(f +g)^{*}(s)\,\mathrm{d}s
\leq \int _{0}^{t}f^{*}(s)\,\mathrm{d}s + \int _{0}^{t}g
^{*}(s)\,\mathrm{d}s.\end{aligned}$$ This inequality can be also written in the form $$\begin{aligned}
\label{E:subadditivity-of-doublestar-a}
(f+g)^{**}\leq f^{**}+g^{**}.\end{aligned}$$ A fundamental result in the theory of Banach function spaces is the *Hardy lemma* [@BS Chapter 2, Proposition 3.6] which states that if two nonnegative measurable functions $f,g$ on $(0,\infty )$ satisfy $$\begin{aligned}
\int _{0}^{t}f(s)\,\mathrm{d}s\leq \int _{0}^{t}g(s)\,\mathrm{d}s\end{aligned}$$ for all $t\in (0,\infty )$, then, for every nonnegative nonincreasing function $h$ on $(0,\infty )$, one has $$\begin{aligned}
\int _{0}^{\infty }f(s)h(s)\,\mathrm{d}s\leq \int _{0}^{\infty }g(s)h(s)\,\mathrm{d}s.\end{aligned}$$

Another important property of rearrangements is the *Hardy-Littlewood inequality* [@BS Chapter 2, Theorem 2.2], which asserts that, if $f, g \in \mathcal{M}(R,\mu )$, then $$\begin{aligned}
\label{E:HL}
\int _{R} |fg| \,\mathrm{d}\mu \leq \int _{0}^{\infty } f^{*}(t) g^{*}(t)\,\mathrm{d}t.\end{aligned}$$

If $(R,\mu )$ and $(S,\nu )$ are two (possibly different) $\sigma $-finite measure spaces, we say that functions $f\in
\mathcal{M}(R,\mu )$ and $g\in \mathcal{M}(S,\nu )$ are *equimeasurable*, and write $f\sim g$, if $f^{*}=g^{*}$ on $(0,\infty )$.

A functional $\varrho \colon \mathcal{M}_{+} (R,\mu ) \to [0,\infty ]$ is called a *Banach function norm* if, for all $f$, $g$ and $\{f_{j}\}_{j\in \mathbb{N}}$ in $\mathcal{M}_{+}(R,\mu )$, and every $\lambda \geq 0$, the following properties hold:

