  We define an integral of real-valued functions with respect to a measure that
takes its values in the extended positive cone of a partially ordered vector
space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated
convergence theorem are established; the analogues of the classical $\mathcal
L^1$- and $L^1$-spaces are investigated. The results extend earlier work by
J.M. Wright and specialise to those for the Lebesgue integral when $E$ equals
the real numbers.
  The hypothesis on $E$ that is needed for the definition of the integral and
for the monotone convergence theorem to hold ($\sigma$-monotone completeness)
is a very mild one. It is satisfied, for example, by the space of regular
operators between a partially ordered vector space with a generating positive
cone and a $\sigma$-monotone complete partially ordered vector space, and by
every JBW-algebra. Fatou's lemma and the dominated convergence theorem hold for
every $\sigma$-Dedekind complete space.
  When $E$ consists of the regular operators on a Banach lattice with an order
continuous norm, or when it consists of the self-adjoint elements of a weakly
closed complex linear subspace of the bounded operators on a complex Hilbert
space, then the finite measures as in the current paper are precisely the
strongly $\sigma$-additive positive operator-valued measures. When $E$ is a
partially ordered Banach space with a closed positive cone, then every positive
vector measure is a measure in our sense, but not conversely. Even when a
measure falls into both categories, the domain of the integral as defined in
this paper can properly contain that of any reasonably defined integral with
respect to the vector measure using Banach space methods.
