Covering theorems and Lebesgue integration

TL;DR

This paper demonstrates how the Lebesgue integral can be derived from Riemann sums and extends the Morse Covering Theorem to open sets, providing a new covering result with explicit bounds.

Contribution

It introduces an extension of the Morse Covering Theorem to open sets and details a method to obtain Lebesgue integrals via Riemann sums using Morse sets.

Findings

Lebesgue integral characterized as a limit of Riemann sums over Morse sets.

Extended Morse Covering Theorem applicable to open sets with explicit bounds.

Provided an estimate for the number of disjoint subfamilies covering the set of Morse set centers.

Abstract

This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let $X$ be a finite dimensional normed space; let $\mu$ be a Radon measure on $X$ and let $\Omega\subseteq X$ be a $\mu$-measurable set. For $\lambda\geq1$, a $\mu $-measurable set $S_{\lambda}(a)\subseteq X$ is a $\lambda$-Morse set with tag $a\in S_{\lambda}(a)$ if there is $r>0$ such that $B(a,r)\subseteq S_{\lambda }(a)\subseteq B(a,\lambda r)$ and $S_{\lambda}(a)$ is starlike with respect to all points in the closed ball $B(a,r)$. Given a gauge $\delta:\Omega \to(0,1]$ we say $S_{\lambda}(a)$ is $\delta$-fine if $B(a,\lambda r)\subseteq B(a,\delta(a))$. If $f\geq0$ is a $\mu$-measurable function on $\Omega$ then $\int_{\Omega}f d\mu=F\in\mathbb{R}$ if and only if for some $\lambda\geq1$ and all $\epsilon>0$ there is a gauge function $\delta$ so that $|\sum_{n}f(x_{n}) \mu(S(x_{n}))-F|<\epsilon$ for all sequences of disjoint $\lambda$-Morse sets that are $\delta$-fine and cover all but a $\mu $-null subset of $\Omega$. This procedure can be applied separately to the positive and negative parts of a real-valued function on $\Omega$. The covering condition $\mu(\Omega\setminus\cup_{n}S(x_{n}))=0$ can be satisfied due to the Morse Covering Theorem. The improved version given here says that for a fixed $\lambda\geq1$, if $A$ is the set of centers of a family of $\lambda$-Morse sets then $A$ can be covered with the interiors of sets from at most $\kappa$ pairwise disjoint subfamilies of the original family; an estimate for $\kappa$ is given in terms of $\lambda$, $X$ and its norm.