We will discuss some topological aspects of the problem which asks to what
extent the vector analogue of the Kalton-Roberts on nearly additive set functions holds true. Namely, we say that a Banach space X has the SVM (stability
of vector measures) property (\(\kappa\)-\(\mathrm{SVM}\)), provided there is a constant \(v(X) < \infty\)
such that for every set algebra \(\mathcal{F}\) (with cardinality less than \(\kappa\)) and every function \(\nu \colon \mathcal{F} \to X\) satisfying
\[\|\nu(A \cup B) - \nu(A) - \nu(B)\| \leq 1 \text{ for all }A,B \in \mathcal{F} \text{ with } A \cap B = \emptyset,\]
there exists a (finitely additive) vector measure \(\mu \colon \mathcal{F} \to X\) satisfying \(\|\mu(A) - \nu(A)\| \leq v(X)\) for each \(A \in \mathcal{F}\).
We will show, e.g., that for compact Hausdorff
spaces \(K\) of finite Cantor-Bendixson height, the Banach space \(C(K)\) has the \(\omega_1\)-\(\mathrm{SVM}\) property and in some cases \(\omega_1\) cannot be improved here.
We will also show how some topological constructions may be used in order to prove that
the Johnson-Lindenstrauss space with index \(p = \infty\), \(\mathrm{JL}_\infty\), has the \(\omega_2\)-\(\mathrm{SVM}\)
property and, again, \(\omega_2\) is the best possible. Several open questions will be also addressed.
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