1.  $\varrho (f)=0$ if and only if $f=0$; $\varrho (\lambda f)= \lambda
    \varrho (f)$; $\varrho (f+g)\leq \varrho (f)+ \varrho (g)$ (the *norm axiom*);

2.  $ f \le g$ a.e. implies $\varrho (f)\le \varrho (g)$ (the *lattice axiom*);

3.  $ f_{j} \nearrow f$ a.e. implies $\varrho (f_{j}) \nearrow \varrho (f)$ (the *Fatou axiom*);

4.  $\varrho (\chi _{E})<\infty $ for every $E\subset R$ of finite measure (the *nontriviality axiom*);

5.  if $E$ is a subset of $R$ of finite measure, then $\int _{E} f\,{\mathrm{d}}\mu \le C_{E} \varrho (f)$ for some positive constant $C_{E}$, depending on $E$ and $\varrho $ but independent of $f$ (the *local embedding in $L^{1}$*).

If, in addition, $\varrho $ satisfies

-   $\varrho (f) = \varrho (g)$ whenever $f^{*} = g^{*}$(the *rearrangement-invariance axiom*),

then we say that $\varrho $ is an *r.i. norm*.

If $\varrho $ is an r.i. norm, then the collection $$\begin{aligned}
X=X({\varrho })=\{f\in \mathcal{M}(R,\mu ): \varrho (|f|)<\infty
\}\end{aligned}$$ is called a *rearrangement-invariant space* (*r.i. space* for short), corresponding to the norm $\varrho $. We shall write $\|f\|_{X}$ instead of $\varrho (|f|)$. Note that the quantity $\|f\|_{X}$ is defined for every $f\in \mathcal{M}(R,
\mu )$, and $$\begin{aligned}
f\in X\quad \Leftrightarrow \quad \|f\|_{X}<\infty .\end{aligned}$$

With any r.i. norm 