Overview
This research presents a constructive proof for the existence of absolutely continuous curves. These curves specifically connect any given pair of measures, denoted as $\mu$ and $\nu$, within the mathematical space $\mathrm{PL}_\infty^p(\mathbb{R}^n)$. This space is defined as the complete metric space of absolutely continuous measures that possess a density in $L^p(\mathbb{R}^n)$ and exhibit bounded support. The proof is established for all values of $p$ such that $1 \leq p \leq \infty$.
Research Context
The investigation focuses on a specific mathematical construct: the space $(\mathrm{PL}_\infty^p(\mathbb{R}^n), \mathfrak{d}_\infty^p)$. This space comprises absolutely continuous measures characterized by a density belonging to the $L^p(\mathbb{R}^n)$ function space and possessing bounded support. The metric $\mathfrak{d}_\infty^p$ defines distances within this space. Earlier work, referenced as [1], introduced this specific complete metric space. The current research builds upon this foundation by addressing connectivity properties within it.
Approach
The core of the research involves a constructive proof. This type of proof not only asserts the existence of a mathematical object (in this case, absolutely continuous curves) but also provides a method or algorithm for its construction. The objective of the construction is to connect any two arbitrary measures, $\mu$ and $\nu$, that are elements of the space $\mathrm{PL}_\infty^p(\mathbb{R}^n)$. The proof is developed for a specified range of the parameter $p$, covering $1\leq p\leq \infty$.
Findings
The central finding is the constructive demonstration of the existence of absolutely continuous curves. These curves are shown to effectively connect every pair of measures $(\mu, \nu)$ from the space $\mathrm{PL}_\infty^p(\mathbb{R}^n)$. This result holds true for all $p$ in the interval $1\leq p\leq \infty$. The space $\mathrm{PL}_\infty^p(\mathbb{R}^n)$ is explicitly identified as the complete metric space of absolutely continuous measures with density in $L^p(\mathbb{R}^n)$ and bounded support.