Order integrals

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