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

\rightarrow(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 $\varepsilon>0$ 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 $\mu(\Omega\setminus\cup_{n}S(x_{n}))=0$ 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.