Ergodic measures in minimal group actions with finite topological sequence entropy

Let $G$ be an infinite discrete countable group and $(X,G)$ be a minimal $G$-system. In this paper, we prove the supremum of topological sequence entropy of $(X,G)$ is not less than $\log ({\sum }_{\mu \in {\mathcal{M}}^e(X,G)}{e}^{h_{\mu }^{\ast }(X,G)})$. If additionally $G$ is abelian then there is a constant $K\in \mathbb{N}\cup \{\mathrm{\infty }\}$ with $\log K\le h_{top}^{\ast }(X,G)$ such that $\nu (\{y\in H:\mathrm{\mid }{\pi }^{-1}(y)\mathrm{\mid }=K\})=1$ where $(H,G)$ is the maximal equicontinuous factor of $(X,G)$, $\pi :(X,G)\to (H,G)$ is the factor map and $\nu$ is the Haar measure of $H$.