Equimeasurable symmetric spaces of measurable function

Abstract

In this paper we consider equimeasurable symmetric(rearrangement invariant) spaces ${\mathbf{E}}_1={\mathbf{E}}_1({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ and ${\mathbf{E}}_2={\mathbf{E}}_2({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ on a measure spaces $({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ and $({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ with finite or infinite $\sigma$-finite non-atomic measures ${\mu }_1$ and ${\mu }_2$. If ${\mathbf{E}}_1({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ be a symmetric space on a measure spaces $({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ and $({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ be a measure space such that ${\mu }_1({\Omega }_1)={\mu }_2({\Omega }_2)$, then there exists a unique symmetric space ${\mathbf{E}}_2({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ on $({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$, which is equimeasurable to ${\mathbf{E}}_1({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$.

22 pages