  A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector
measures) property if there exists a constant $v<\infty$ such that for any
algebra of sets $\mathcal F$, and any function $\nu\colon\mathcal F\to X$
satisfying $$\|\nu(A\cup B)-\nu(A)-\nu(B)\|\leq 1\quad{for
disjoint}A,B\in\mathcal F,$$there is a vector measure $\mu\colon\mathcal F\to
X$ with $\|\nu(A)-\mu(A)\|\leq v$ for all $A\in\mathcal F$. 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
$\kappa$-$\mathsf{SVM}$ property. The least cardinal $\kappa$ for which $X$
does not have the $\kappa$-$\mathsf{SVM}$ property (if it exists) is called the
$\mathsf{SVM}$ character of $X$. We apply the machinery of twisted sums and
quasi-linear maps to characterise these properties and to determine
$\mathsf{SVM}$ characters for many classical Banach spaces. We also discuss
connections between the $\kappa$-$\mathsf{SVM}$ property, $\kappa$-injectivity
and the `three-space' problem.
