  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.
