del Campo, Ricardo; Sánchez-Pérez, Enrique A. Positive representations of \(L^{1}\) of a vector measure. (English) Zbl 1137.46028 Positivity 11, No. 3, 449-459 (2007). For a positive measure \(\mu\) and \(f\in L_1 (\mu)\), it is elementary that \(\| f\| =\int | f| \:d\mu\). If \(\mu\) has its values in a Banach lattice, it is called positive if its range is a subset of the positive cone \(C_X^+\). In this case, we have \(\| f\| =\| \int | f| \:d\mu\| _X\). The integral \(\int f\:d\mu\) is to be considered as the Bartle-Dunford-Schwarz integral with respect to the countably additive \(X\)-valued measure \(\mu\). If \(\mu\) is no longer positive, it could still be true that the formula above gives an equivalent norm (or quasi-norm) on \(L_1 (\mu)\). In Section 2, it is proved that the formula gives an equivalent quasi-norm if and only if \(\mu:\Sigma\rightarrow X\) is a cone-open transformation of a positive measure, that is, there are a Banach lattice \(Z\), a positive measure \(\nu:\Sigma\rightarrow Z\) and a cone-open operator \(S:Z\rightarrow X\) such that \(\mu=S\circ \nu\). If the formula gives an equivalent (quasi-) norm on \(L_1 (\mu)\), one may wonder whether \(X\) can be re-ordered in some way such that \(\mu\) becomes positive with respect to this new order structure. A measure which becomes positive under a re-ordering that turns \(X\) into an \(M\)-normed Riesz space is called pseudo-positive. In Section 3, it is proved that a measure is pseudo-positive if and only if it is a cone-open and lattice-generating transformation of a positive measure. In particular, thus, for pseudo-positive measures \(\mu\), \(\| \int | f| \:d\mu\| _X\) gives an equivalent quasi-norm for \(L_1(\mu)\). Reviewer: Olav Nygaard (Kristiansand) Cited in 1 Document MSC: 46G10 Vector-valued measures and integration 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:positive vector measure; equivalent quasinorm; cone-open operator PDFBibTeX XMLCite \textit{R. del Campo} and \textit{E. A. Sánchez-Pérez}, Positivity 11, No. 3, 449--459 (2007; Zbl 1137.46028) Full Text: DOI References: [1] C. D. Aliprantis, K. C. Border, Infinite dimensional analysis Springer, Berlin (1999). [2] R. G. Bartle, N. Dunford, J. Schwartz, Weak compactness and vector measures. Can. J. Math. 7 (1955), 289–305. · Zbl 0068.09301 [3] G. P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330. · Zbl 0735.46033 [4] J. Diestel, J. J. Uhl Jr., Vector measures. Math. Surveys 15, Am. Math. Soc. Providence (1977). · Zbl 0369.46039 [5] A. Fernández, F. Mayoral, F. Naranjo, C. Sáez, E. A. Sánchez Pérez, Spaces of integrable functions with respect to a vector measure and factorizations through Lp and Hilbert spaces. J. Math. Anal. Appl. 330 (2007), 1249–1263 · Zbl 1129.47019 [6] D. R. Lewis, Integration with respect to vector measures. Pacific J. Math. 23(1) (1970), 157–165. · Zbl 0195.14303 [7] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II. Springer. Berlin (1996). · Zbl 0852.46015 [8] W. A. J. Luxemburg, A. C. Zaanen, Riesz spaces. North-Holland, Vol I, Amsterdam (1971). · Zbl 0231.46014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.