Mathematics offers various notions of "generalized measures." In this talk, I will focus on two key examples from topology: standard Borel measures and the ideal-valued measures introduced by Gromov. Each has a symplectic analogue, which plays a central role in many results in symplectic topology. I will explain how these symplectic generalized measures relate to their topological counterparts via the Poisson bracket, and describe applications of ideal-valued measures in symplectic rigidity. This talk is based on joint works with Yaniv Ganor, Leonid Polterovich and Frol Zapolsky.