# Article

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Keywords:
liftings; lifting topology; weakly compact sets; Radon-Nikodym derivative
Summary:
For a Banach space $E$ and a probability space $(X, \mathcal{A}, \lambda)$, a new proof is given that a measure $\mu: \mathcal{A} \to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C \subset E$ such that $|\mu |_{C} : \mathcal{A} \to [0, \infty]$ is a finite valued countably additive measure. Here we define $|\mu |_{C}(A) = \sup \{\sum_{k} |\langle \mu (A_{k}), f_{k}\rangle |\}$ where $\{A_{k}\}$ is a finite disjoint collection of elements from $\mathcal{A}$, each contained in $A$, and $\{f_{k}\}\subset E'$ satisfies $\sup_{k} |f_{k} (C)|\leq 1$. Then the result is extended to the case when $E$ is a Frechet space.
References:
[1] Davis W.J., Figiel T., Johnson W.B., Pelczynski A.: Factoring weakly compact operators. J. Funct. Anal. 17 (1974), 311–327. DOI 10.1016/0022-1236(74)90044-5 | MR 0355536 | Zbl 0306.46020
[2] Diestel J., Uhl J.J.: Vector Measures. Amer. Math. Soc. Surveys, 15, American Mathematical Society, Providence, RI, 1977. MR 0453964 | Zbl 0521.46035
[3] Gruenwald M.E., Wheeler R.F.: A strict representation of $L_{1}(\mu, X)$. J. Math. Anal. Appl. 155 (1991), 140–155. MR 1089331
[4] Khurana S.S.: Topologies on spaces of continuous vector-valued functions. Trans Amer. Math. Soc. 241 (1978), 195–211. DOI 10.1090/S0002-9947-1978-0492297-X | MR 0492297
[5] Khurana S.S.: Topologies on spaces of continuous vector-valued functions II. Math. Ann. 234 (1978), 159–166. DOI 10.1007/BF01420966 | MR 0494178
[6] Khurana S.S.: Pointwise compactness and measurability. Pacific J. Math. 83 (1979), 387–391. DOI 10.2140/pjm.1979.83.387 | MR 0557940 | Zbl 0425.46009
[7] Phelps R.R.: Lectures on Choquet's Theorem. D. van Nostrand Company, Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470 | Zbl 0997.46005
[8] Schaefer H.H.: Topological Vector Spaces. Springer, 1986. MR 0342978 | Zbl 0983.46002
[9] Ionescu Tulcea A., Ionescu Tulcea C.: Topics in the theory of lifting. Springer, New York, 1969. MR 0276438 | Zbl 0179.46303

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