The sum of two Radon-Nikodým-sets need not be a Radon-Nikodým-set

Schachermayer, Walter

doi:10.1090/S0002-9939-1985-0796445-2

The sum of two Radon-Nikodým-sets need not be a Radon-Nikodým-set
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Proc. Amer. Math. Soc. 95 (1985), 51-57 Request permission

Abstract:

It was shown by C. Stegall that, if $C$ is a Radon-Nikodym-set and $K$ weakly compact, then $K + C$ is a Radon-Nikodym-set. We show that there are closed, bounded, convex Radon-Nikodym-sets ${C_1}$ and ${C_2}$ such that ${C_1} + {C_2}$ is closed but contains an isometric copy of the unit-ball of ${c_0}$. In fact, we give two examples, one following the lines of one due to McCartney and O’Brian, the other due to Bourgain and Delbaen. We also give an easy example of a non-Radon-Nikodym-set $C$ such that, for every $\varepsilon > 0$, there is a Radon-Nikodym-set ${C_\varepsilon }$ such that $C$ is contained in the sum of ${C_\varepsilon }$ and the ball of radius $\varepsilon$.

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