The decreasing rearrangements of functions for vector-valued measures

Abstract

Let $B$ be a complete Boolean algebra, let $Q(B)$ be the Stone compact of $B$, let $C_\infty (Q(B))$  be the commutative unital algebra of all continuous functions  $x:Q(B) \rightarrow [-\infty,+\infty],$ assuming possibly the  values $\pm\infty$  on nowhere-dense subsets of $Q(B)$. We consider Maharam measure $m$ defined on $B$, which takes on value in the algebra $L^0(\Omega)$ of all real measurable functions on the measurable space  $(\Omega, \Sigma, \mu)$ with a $\sigma$-finite numerical measure  $\mu$.  The decreasing rearrangements  of functions from $C_\infty (Q(B))$, associated with such a measure $m$ and taking values in the algebra $L^0(\Omega)$ are determined. The basic properties of such rearrangements are established, which are similar to the properties of classical decreasing rearrangements  of measurable functions.
As an application, with the help of the property of equimeasurablity of elements from $ C_\infty (Q(B))$, associated with such a measure $m$, the notion of a symmetric Banach-Kantorovich space  $(E,\|\cdot\|_{E})$ over $L^0(\Omega)$ is introduced and studied in detail. Here $E\subset C_\infty (Q(B)),$ and \ $\|\cdot\|_{E}$ -- $L^0(\Omega)$-valued norm in $E$, endowing it with the structure of the Banach-Kantorovich space. Examples of symmetric Banach-Kantorovich spaces are given, which are vector-valued analogues of classical $L^p$-spaces, $ 1\leq p \leq \infty$, associated with a numerical $\sigma$-finite measure.