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Journal of Convex Analysis 15 (2008), No. 2, 313--324
Copyright Heldermann Verlag 2008

Preservation of the Range of a Vector Measure under Shortenings of the Domain

Dept. of Mathematics, University of Delhi, Delhi 110007, India
Dept. of Mathematics, Mizoram University, Aizawl 796001, Mizoram, India

Ajit Iqbal Singh
Dept. of Mathematics, University of Delhi, South Campus, Benito Juarez Road, New Delhi 110021, India


Let $\bm{\mu}$ be a non-zero, non-atomic vector measure on the measurable space $(\bm{X}, \bm{\Omega})$ taking values in $\bm{R}^n$. Liapounov's convexity theorem gives that the range $R(\mu)$ is convex, an immediate consequence of this is that there exist uncountably many smaller collections $S\subset \Omega$ with preservation of the range, that is $\bm{R}(\bm{\mu}) = \bm{R}(\bm{\mu} / S)$. In case $\bm{X}$ is a topological space and $\bm{\Omega}$ the class of Borel sets, such reductions consisting of open sets or other sets related to continuous functions on $\bm{X}$ have been obtained by H. Render and H. Stroetmann [Arch. Math. 67 (1996) 331--336], D. Wulbert [Proc. Amer. Math. Soc. 108 (1990) 955--960; Proc. Amer. Math. Soc. 128 (2000) 2431--2438; Israel J. Math. 126 (2001) 363-380; J. Functional Analysis 182 (2001) 1-15] and H. G. Kellerer [Arch. Math. 72 (1999) 206--213; Proc. Amer. Math. Soc. 130 (2002) 2305--2309]. J. M. Gouweleeuw [Indag. Math. N.S. 4 (1993) 141--161; Proc. London Math. Soc., III. Ser. 70 (1995) 336--362] gave a decomposition of an $\bm{R}^n$-valued vector measure $\bm{\mu} = (\mu_1, \mu_2, \ldots, \mu_n)$ on the measurable space $(\bm{X}, \bm{\Omega})$, where each $\mu_i$ is a non-negative real measure on $(\bm{X}, \bm{\Omega})$, based on the atoms of the measure $\bm{\mu}$. She also characterized those $\bm{\mu}$ which have a convex range. J. van Mill and A. Ran [Indag. Mathem., N.S. 7 (1996) 227--242] gave various interesting variants and generalizations of the Gouweleeuw decomposition and convexity results. It is the purpose of this paper to apply these decompositions and make attempts to shorten the domain $\bm{\Omega}$ to various minimal subsets of $\bm{\Omega}$ called shortenings, which preserve the range of $\bm{\mu}$. We also make use of the R\'enyi criteria and the work on interval filling sequences of Z. Dar\'oczy, A. J\'arai, I. Kat\'ai and T. Szab\'o.

Keywords: Vector measure, preservation of the range, shortening of the domain, Gouweleeuw decomposition, van Mill and Ran decomposition, interval filling sequence.

MSC: 28B05; 26A51, 46G10, 52A05

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