Satisfaction Problem of Consumers Demands measured by ordinary "Lebesgue measures" in $R^{\infty}$

TL;DR

This paper investigates the minimal initial measure needed to satisfy consumer demands in an infinite-dimensional space under various dynamical systems, using Lebesgue measures and Liouville type theorems.

Contribution

It introduces a framework for solving the Satisfaction Problem of Consumers Demands in infinite-dimensional spaces using measure theory and dynamical systems analysis.

Findings

Derived relations between initial and final measures using Liouville theorems.

Provided solutions for minimal initial measures in various dynamical systems.

Extended measure satisfaction analysis to infinite-dimensional settings.

Abstract

In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): {\it The supplier must supply the measurable system of the measure $m_k$ to the $k$-th consumer at time $t_k$ for $1 \le k \le n$. The measure of the supplied measurable system is changed under action of some dynamical system, What is a minimal measure of measurable system which must take the supplier at the initial time $t=0$ to satisfy demands of all consumers ?} In this paper we consider Satisfaction Problem of Consumers Demands measured by ordinary "Lebesgue measures" in $R^{\infty}$ for various dynamical systems in $R^{\infty}$. In order to solve this problem we use Liouville type theorems for them which describes the dependence between initial and resulting measures of the entire system.