Stability of vector measures and twisted sums of Banach spaces

A Banach space $X$ is said to have the $\mathsf{S}\mathsf{V}\mathsf{M}$ (stability of vector measures) property if there exists a constant $v<\mathrm{\infty }$ such that for any algebra of sets $\mathcal{F}$, and any function $\nu \text{\hspace{0.17em}⁣}:\mathcal{F}\to X$ satisfying $$\mathrm{\parallel }\nu (A\cup B)-\nu (A)-\nu (B)\mathrm{\parallel }\le 1fordisjointA,B\in \mathcal{F},$$there is a vector measure $\mu\colon\mathcal F\to X$with$|\nu(A)-\mu(A)|\leq v$forall$A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal{F}$ of cardinality less than some fixed cardinal number $\kappa$, then we say that $X$ has the $\kappa$-$\mathsf{S}\mathsf{V}\mathsf{M}$ property. The least cardinal $\kappa$ for which $X$ does not have the $\kappa$-$\mathsf{S}\mathsf{V}\mathsf{M}$ property (if it exists) is called the $\mathsf{S}\mathsf{V}\mathsf{M}$ character of $X$. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine $\mathsf{S}\mathsf{V}\mathsf{M}$ characters for many classical Banach spaces. We also discuss connections between the $\kappa$-$\mathsf{S}\mathsf{V}\mathsf{M}$ property, $\kappa$-injectivity and the `three-space’ problem.