A vector measure with no derivative

Lewis, D. R.

doi:10.1090/S0002-9939-1972-0296248-2

A vector measure with no derivative

Author: D. R. Lewis
Journal: Proc. Amer. Math. Soc. 32 (1972), 535-536
MSC: Primary 28A45
DOI: https://doi.org/10.1090/S0002-9939-1972-0296248-2
MathSciNet review: 0296248
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a nonatomic scalar measure $\mu$ , there is a vector valued, $\mu$ -continuous measure of finite variation which has no derivative with respect to $\mu$ , but which has the property that the closure of its range is compact and convex.

[1] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 0115069, https://doi.org/10.4064/sm-17-2-151-164
[2] Nelson Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323–392. MR 0002020, https://doi.org/10.1090/S0002-9947-1940-0002020-4
[3] A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648. MR 0149295
[4] D. R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to ${l_1}(\Gamma )$ (submitted).
[5] J. J. Uhl Jr., The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163. MR 0264029, https://doi.org/10.1090/S0002-9939-1969-0264029-1