Covering theorems and Lebesgue integration
Abstract
Description
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 $μ$ be a Radon measure on $X$ and let $Ω\subseteq X$ be a $μ$-measurable set. For $λ\geq1$, a $μ$-measurable set $S_λ(a)\subseteq X$ is a $λ$-Morse set with tag $a\in S_λ(a)$ if there is $r>0$ such that $B(a,r)\subseteq S_{λ}(a)\subseteq B(a,λr)$ and $S_λ(a)$ is starlike with respect to all points in the closed ball $B(a,r)$. Given a gauge $δ:Ω\to(0,1]$ we say $S_λ(a)$ is $δ$-fine if $B(a,λr)\subseteq B(a,δ(a))$. If $f\geq0$ is a $μ$-measurable function on $Ω$ then $\int_Ωf dμ=F\in\mathbb{R}$ if and only if for some $λ\geq1$ and all $ε>0$ there is a gauge function $δ$ so that $|\sum_{n}f(x_{n}) μ(S(x_{n}))-F|<ε$ for all sequences of disjoint $λ$-Morse sets that are $δ$-fine and cover all but a $μ$-null subset of $Ω$. This procedure can be applied separately to the positive and negative parts of a real-valued function on $Ω$. The covering condition $μ(Ω\setminus\cup_{n}S(x_{n}))=0$ can be satisfied due to the Morse Covering Theorem. The improved version given here says that for a fixed $λ\geq1$, if $A$ is the set of centers of a family of $λ$-Morse sets then $A$ can be covered with the interiors of sets from at most $κ$ pairwise disjoint subfamilies of the original family; an estimate for $κ$ is given in terms of $λ$, $X$ and its norm.
To appear in Mathematica Japonica, 13 pages
To appear in Mathematica Japonica, 13 pages