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A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $\left(a_n\right)$, $\left(b_m\right)$ in $𝒜$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in $𝒜$ such that $a_n\le b\le b_n$ for all $n$. Let $𝒜$ denote an algebra with the property (I). It is shown that if $({\mu }_n:𝒜\to X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_n{\mu }_n\left(a\right)$ exists for each $a\in 𝒜$, then $\mu \left(a\right)={lim}_n{\mu }_n\left(a\right)$ defines a strongly additive map from $𝒜$ to $X$ the ${\mu }_n^{\text{'}}s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued...