Liapounov's convexity theorem for topological measures (Q1924986)

Let \(\mu_i\) be non-atomic Borel measures on a separable metric space for \(i = 1,\dots, n\). Then the restricted range \(\{(\mu_1(U),\dots,\mu_n(U))\): \(U\) open\} is a convex set which is in general not closed. On the other side, the range is closed if the measures \(\mu_i\), defined over an open or closed interval, possess real-analytic densities for an appropriate dominating non-atomic measure \(\nu\). If the interval is compact there exists a natural number \(N\) such that for each Borel set \(B\) there exist intervals \(I_1,\dots, I_m\), \((m \leq N)\) such that \(\mu_i(B) = \mu_i(\bigcup^m_{k = 1} I_k)\) for all \(i = 1,\dots,n\).