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Geometric properties of the range of two-dimensional quasi-measures with respect to the Radon-Nikodým property. (English) Zbl 0771.28002
Let $$\mu$$ and $$\nu$$ be nonnegative, finite quasi-measures (finitely additive measures) on a $$\sigma$$-field $${\mathcal A}$$ of subsets of a set $$\Omega$$. In addition, assume that $$\mu$$ and $$\nu$$ are continuous in the sense that for each $$\varepsilon>0$$ there is a finite partition $$\{A_ 1,A_ 2,\dots,A_ n\}$$ of $$\Omega$$ such that $$A_ i\in{\mathcal A}$$ and $$\mu(A_ i)<\varepsilon$$ for each $$i$$, and similarly for $$\nu$$. This paper investigates existence of a Radon-Nikodým derivative of $$\nu$$ with respect to $$\mu$$, that is, a nonnegative integrable function $$f$$ on $$\Omega$$ for which $$\nu(E)=\int_ Ef d\mu$$ for all $$E\in{\mathcal A}$$, from the perspective of the range $$R=\{(\mu(E),\nu(E)):E\in{\mathcal A}\}$$. When such a function $$f$$ exists, $$(\mu,\nu)$$ is called an integral quasi-measure.
In this setting, there are nondecreasing functions $$G$$ and $$g$$, defined on the range of $$\mu$$, which form the upper and lower boundaries of $$R$$. The upper function $$G$$ is concave while the lower function $$g$$ is convex. Assuming $$\nu\ll\mu$$, it is shown that “$$R$$ is closed” is a sufficient but not necessary condition for $$(\mu,\nu)$$ to be integral. More generally, a condition on which points of the boundary lie in $$R$$ is shown to be necessary and sufficient for $$(\mu,\nu)$$ to be integral. Finally, it is shown that any concave, nondecreasing function $$G:[0,a]\to[0,\infty)$$ with $$G(0)=0$$ determines the upper boundary of the range of an integral quasi-measure.

##### MSC:
 28A15 Abstract differentiation theory, differentiation of set functions
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##### References:
 [1] Armstronǵ, T.E; Prikry, K, Liapounoff’s theorem for non-atomic, finitely additive, bounded, finite-dimensional, vector-valued measures, Trans. amer. math. soc., 266, 499-515, (1981) · Zbl 0518.28005 [2] Bolker, E.D, A class of convex bodies, Trans. amer. math. soc., 145, 323-345, (1969) · Zbl 0194.23102 [3] Candeloro, D; Martellotti, A, Su alcuni problemi relativi a misure scalari subadditive, e applicazioni al caso dell′additività finita, Atti sem. mat. fis. univ. modena, 27, 284-296, (1978) · Zbl 0427.28003 [4] Candeloro, D; Martellotti, A, Sul rango di una massa vettoriale, Atti sem. mat. fis. univ. modena, 28, 102-111, (1979) · Zbl 0438.28008 [5] Greco, G.H, Un teorema di Radon-Nikodym per funzioni d′insieme subadditive, Ann. univ. ferrara seź. VII, 27, 13-19, (1981) [6] Khurana, S.S, A note on Radon-Nikodym theorem for finitely additive measures, Pacific J. math., 74, 103-104, (1978) · Zbl 0371.28005 [7] Martellotti, A, Countably additive restrictions of vector-valued quasi-measures with respect to range preservation, Bollettino U.M.I., 2-B, 7, 445-458, (1988) · Zbl 0647.28005 [8] Maynard, H.B, A Radon-Nikodym theorem for finitely additive bounded measures, Pacific J. math., 83, 401-413, (1979) · Zbl 0453.28004
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