Abstract: Let μ₁, μ₂, ..., μ_n be non-atomic probability measures on a measurable space (X, Σ).

Theorem (Liapanouv 1940) {μ(U) = (μ₁(U), μ₂(U) , ... , μ_m(U) ): U in Σ} is a compact convex set.

If in addition there is a topology on X and Σ is the Borel sets (or the Baire sets, respectively) we can ask when the range of the vector valued measure μ is obtained even when the measure is restricted to the sets U which are open (or the support of a non-negative continuous function, resp.).

We will give a couple of 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 Σ be the sigma-algebra generated by S. The three cases above result when L = l_∞, the upper-semi-continuous functions on X, and C(X) respectively.