Equimeasurable symmetric spaces of measurable function
In this paper we consider equimeasurable symmetric(rearrangement invariant) spaces $\mathbf{E}_1 = \mathbf{E}_1(Ω_1,\mathcal{F}_1,μ_1)$ and $\mathbf{E}_2 = \mathbf{E}_2(Ω_2,\mathcal{F}_2,μ_2)$ on a measure spaces $(Ω_1, \mathcal{F}_1,μ_1)$ and $(Ω_2,\mathcal{F}_2,μ_2)$ with finite or infinite $σ$-finite non-atomic measures $μ_1$ and $μ_2$. If $\mathbf{E}_1(Ω_1,\mathcal{F}_1,μ_1)$ be a symmetric space on a measure spaces $(Ω_1, \mathcal{F}_1,μ_1)$ and $(Ω_2,\mathcal{F}_2,μ_2)$ be a measure space such that $μ_1 (Ω_1)=μ_2(Ω_2)$, then there exists a unique symmetric space $\mathbf{E}_2(Ω_2,\mathcal{F}_2,μ_2)$ on $(Ω_2,\mathcal{F}_2,μ_2)$, which is equimeasurable to $ \mathbf{E}_1(Ω_1,\mathcal{F}_1,μ_1)$.
math.FA↗