Uniformly countably additive families of measures and group invariant measures (Q1817886)

Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(\Omega\) and \(\mu\) a \(\sigma\)-additive probability measure on \(\Sigma\). Let \(T\) be a group of permutations on \(\Omega\) such that \(\tau E\in\Sigma\) for all \(\tau\in T\) and all \(E\in\Sigma\). Moreover, it is assumed that \(\{\mu\circ\tau: \tau\in T\}\) is uniformly \(\sigma\)-additive. It is proved that under certain conditions on \(T\) there is a \(\sigma\)-additive probability measure \(\nu\) on \(\Sigma\) such that \(\nu\) is invariant under \(T\) and \(0\leq \nu(E)\leq \sup_{\tau\in T}\mu(\tau^{-1}(E))\) for all \(E\in\Sigma\).