Covering theorems and Lebesgue integration (Q2731707)

From authors' 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 \(\lambda\geq 1\), 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 \(f\geq 0\) is a \(\mu\)-measurable function on \(\Omega\) then \(\int_\Omega f d\mu= F\in{\mathbf R}\) if and only if for some \(\lambda \geq 1\) and all \(\varepsilon> 0\) there is a gauge function \(\delta\) so that \(|\sum_n f(x_n) \mu(S(x_n))- F|< \varepsilon\) 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 \(\mu(\Omega\setminus\bigcup_n S(x_n))= 0\) can be satisfied due to the Morse Covering Theorem. The improved version given here says that for a fixed \(\lambda\geq 1\), 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''.