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Monotone Convergence Theorem; Kurzweil vector integral; ordered normed spaces
We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, $\int _R{\mathrm d}\alpha (t) f(t)$, where $R$ is a compact interval of $\mathbb{R}^n$, $\alpha $ and $f$ are functions with values on $L(Z,W)$ and $Z$ respectively, and $Z$ and $W$ are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, $\int _R\alpha (t)\mathrm{d}f(t)$, as well as to unbounded intervals $R$.
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