let $\mu_1, \mu_2, ... , \mu_m$ be non-atomic probability measures on a measurable space $(x, \sigma)$. theorem (liapanouv 1940) ${\mu(\cup) = (\mu_1(\cup) + \mu_2(\cup) + ... + \mu_m(\cup) ): \cup \ {\text{in}}\ \sigma }$ is a compact convex set. if in addition there is a topology on x and ‚$\sigma$ is the borel sets (or the baire sets, respectively) we can ask when the range of the vector valued measure $\mu$ is obtained even when the measure is restricted to the sets $\cup$ which are open (or the support of a non-negative continuous function, resp.). we will give a couple applications of the classical theorem. we will then cast the liapanouv theorem in an equivalent form about the range of a vector of integrals on x. in that form we will give a single theorem that, in addition to proving the classical liapanouv theorem, also characterizes when the open sets (or the supports of continuous functions, resp.) suffice. that is let l be a cone of functions. let s be the supports of functions in l, and let ‚$\sigma$ be the sigma-algebra generated by s. the three cases above result when $l = l\infty$, the upper-semi-continuous functions on x, and c(x) respectively.