Note on linearity of rearrangement-invariant spaces (Q259021)

Let \(({\mathcal R},{\mathcal X},\mu)\) be a non-atomic \(\sigma\)-finite measure space and let \({\mathcal M}({\mathcal R})\) denote the set of real-valued \(\mu\)-measurable functions. The distribution function \(f_0\) and non-increasing rearrangement \(f^*\) of \(f\) are defined, whenever the terms are finite, by \[ f_0(s)= \mu(\{x: |f(x)|> s\}),\;f^*(t)= \inf\{s> 0: f_0(s)\leq t\}. \] If \(\|\cdot\|_X:{\mathcal M}({\mathcal R})\to (0,\infty)\) is a functional and \(X\) is defined to be \(\{f\in{\mathcal M}({\mathcal R}):\| f\|_X<\infty\}\), then \(X\) is said to be a rearrangement invariant lattice if (i.1) \(f^*= g^*\) implies that \(\| f\|_X=\| g\|_X\), (i.2) \(|f|\leq |g|\) \(\mu\)-a.e. implies that \(\| f\|_X\leq\| g\|_X\), (i.3) \(\| af\|_X= |a|\,\| f\|_X\). If there is a functional \(\|.\|_{\dot X}\) such that \(\| f\|_X=\| f^*\|_{\dot X}\), then \(\|.\|_{\dot X}\) is said to be a representation functional of \(X\). The main theorem of this paper includes the following statements: If \(X\) is a rearrangement invariant lattice, there is a representation functional \(\|\cdot\|_{\dot X}\) for \(X\), and \(\dot X= \{f :\| f\|_{\dot X}<\infty\}\) is a linear space, then \(X\) is a linear space if and only if \(\| f^*\|_X<\infty\) implies that \(\| E_2f^*\|_{\dot X}<\infty\), where \(E_2g(x)= g(x/2)\). In addition, if \(\|\cdot\|_{\dot X}\) is a norm, then \(\|\cdot\|_X\) is a norm if and only if \(\| E_2 f^*\|_{\dot X}\leq 2\| f^*\|_{\dot X}\).