The Mackey continuity of the monotone rearrangement

Abstract:

Let $(A, \mathcal {A},\mu )$ be a probability space, and let mes denote the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal {B}$ in $[0,1]$. The nondecreasing-rearrangement operator from the space ${L^\infty }(\mu ) = {L^\infty }(A, \mathcal {A}, \mu )$ of real-valued essentially bounded functions into ${L^\infty } = {L^\infty }([0,1]$, $\mathcal {B}$, mes) is shown to be uniformly continuous in the Mackey topologies $\tau ({L^\infty }(\mu )$, ${L^1}(\mu ))$ and $\tau ({L^\infty },{L^1})$ on ${L^\infty }(\mu )$ and ${L^\infty }$, respectively.