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.
| Mualliflar | Chilin , V.I., Zakirova , G.B. |
|---|---|
| Jurnal | O‘zbekiston matematika jurnali |
| Nashr sanasi | 2025-11-03 |
| Jild | 69 |
| Son | 4 |
| Betlar | 70-82 |
| Til | Ingliz |
| DOI | 10.29229/uzmj.2025-4-8 |
DOI: 10.29229/uzmj.2025-4-8 · Maqolaning asl sahifasi
vector integration, vector-valued measure, decreasing rearrangements, equimeasurablity, the Banach-Kantorovich lattice, symmetric space
In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a…
Let $Q$ be the field of rational numbers and $Q^{2}$ be the $2$-dimensional linear space over $Q$. A classification of all non-degenerate symmetric bilinear-metric forms over $Q^{2}$ have obtained. Let $\varphi $ be a…
Hypergeometric functions are divided into complete and confluent functions. For the first time, Srivastava and Karlsson described a method for constructing a set of all possible triple Gaussian hypergeometric series and…
Let $R$ be a ring, and let $M$ be a right $R$-module. Recently, Baannon and Khalid introduced and studied the concept of $e^*$-essential submodules, extending the notion of essential submodules. In this context…
In the paper, we consider a class dynamic control system with a discrete parameter and under conditions of uncertainty in the initial data. The optimal control problem of the minimax type is formulated for a non-smooth…
In this paper, we classify the symmetric Leibniz algebras whose underlying Lie algebra is almost filiform. We also describe the symmetric Leibniz algebras associated with Heisenberg and triangular Lie algebras…
We analyze an evasion differential game involving one evader and multiple pursuers in $\R^n$. The control functions of the players are subject to exponential integral constraints to ensure bounded energy consumption…
The time-optimal problem is considered for the controllable heat conduction process in a rod. Using the Fourier method, the problem is usually reduced to an infinite system of one-dimensional control equations whose…
This paper studies an analog of the Tricomi problem for a loaded parabolic-hyperbolic equation of the second kind, which degenerates inside the domain. In proving the existence and uniqueness theorem for a classical…
In this paper, we first classify all associative algebra structures on a two-dimensional vector space over an arbitrary base field equipped with a non-degenerate bilinear form. We then determine which of these are…