Measures provide valuable insights into long-term and global behaviors across various dynamical systems. In this talk, I present my research group's recent works on employing measure theory and optimal transport, combined with powerful tools from modern machine learning, to tackle challenges in dynamical system modeling. The research advances include using the invariant measure for dynamical system parameter identification, extending the celebrated Takens' embedding from the state space to probability space, and proposing the distributional Koopman operator framework to handle variance prediction for stochastic dynamical systems. These works demonstrate the excellent research potential of measure-theoretic approaches for dynamical systems.