[Paper Review] Pure Measures, Density Measures and the Dual of L-infinity

Moritz Schönherr, Friedemann Schuricht|arXiv (Cornell University)|Oct 5, 2017

Advanced Numerical Analysis Techniques7 references3 citations

TL;DR

This paper introduces density measures as a new class of finitely additive measures that provide a concrete, meaningful representation of the dual space of $L^∞(\mathbb{R}^n)$, extending beyond abstract constructions. It establishes that such measures are pure and can be used to represent traces and surface integrals, offering a novel framework for differential calculus and functional analysis on $\mathbb{R}^n$.

ABSTRACT

Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The main results presented here are a more precise representation of the dual of the space of essentially bounded functions, leading to the notion of pure measures, and the definition and analysis of density measures which constitute a large class of such measures. It is shown that density measures have applications in the context of traces. In particular, new and meaningful examples of pure measures are given on Rn, in contrast to common examples in the literature, which are usually constructed on N.

Motivation & Objective

• To refine the representation of the dual space of $L^\infty(\Omega, \mathcal{L}^n)$ beyond general bounded additive measures.
• To define and analyze density measures as a large, meaningful class of pure measures on $\mathbb{R}^n$.
• To provide concrete examples of pure measures in $\mathbb{R}^n$, contrasting with typical constructions on $\mathbb{N}$.
• To demonstrate applications in trace theory and differential calculus using these measures.
• To characterize extremal points of the set of density measures and link them to singular Radon measures.

Proposed method

• Uses lattice-theoretic decomposition techniques to analyze vector lattices of measures and identify pure components.
• Defines density measures via integration against Radon measures supported on sets of Lebesgue measure zero, such as boundaries or lower-dimensional sets.
• Applies core measure theory and duality to show that density measures represent elements of $(L^\infty)^*$.
• Employs the concept of core of a measure to identify the support of pure measures, particularly on sets like $\partial\Omega$.
• Utilizes the Riesz representation theorem and its extensions to link integration against $\mu$ to classical surface integrals.
• Applies the Gauss-Green formula to show that surface fluxes can be represented as integrals with respect to pure, density-type measures.

Experimental results

Research questions

• RQ1 How can the dual space of $L^\infty(\Omega, \mathcal{L}^n)$ be refined beyond general bounded additive measures?
• RQ2 What is the structure of pure measures in $L^\infty(\mathbb{R}^n)^*$, and how can they be explicitly constructed on $\mathbb{R}^n$?
• RQ3 Can density measures serve as a meaningful class of pure measures with applications in trace theory?
• RQ4 What is the role of the core of a measure in characterizing pure measures and their supports?
• RQ5 How do density measures relate to classical surface integrals and differential calculus?

Key findings

• Density measures are a large class of pure measures in $(L^\infty(\Omega, \mathcal{L}^n))^*$, defined via integration against Radon measures on sets of Lebesgue measure zero.
• For any closed set $C \subset \overline{\Omega}$ with $\mathcal{L}^n(C \cap \Omega) = 0$ and positive measure in every neighborhood, a pure measure $\mu$ exists such that $\int_\Omega \phi \, d\mu = \int_C \phi \, d\sigma$ for all $\phi \in C_0(\Omega)$.
• The measure $\mu$ is pure if the associated Radon measure $\sigma$ is singular with respect to Lebesgue measure.
• The core of such a measure $\mu$ is contained in the support set $C$, and $\mu$ can be represented as a surface integral via $\int_\Omega \phi \, d\mu = \int_{\partial\Omega} \phi \, d\mathcal{H}^{n-1}$ for $\Omega$ with smooth boundary.
• The surface flux $\int_{\partial\Omega} \phi \cdot \nu \, d\mathcal{H}^{n-1}$ can be represented as $\int_\Omega \phi \, d\mu$ for a pure measure $\mu$, linking it to the divergence theorem.
• Extremal points of the set of density measures are characterized as those corresponding to Dirac-type measures on sets of zero Lebesgue measure.