In many areas of mathematics there are theorems of the following kind: Any map in a suitable class has a big fiber. The classes of maps and the notions of size vary from field to field. In my talk I'll present several examples of this phenomenon. I'll show how Gromov's notion of ideal valued-measures derived from cohomology can be used to prove some of them. I'll also introduce objects which are a suitable generalization of ideal-valued measures in the context of symplectic geometry, called ideal-valued quasi-measures, indicate how they can be constructed using relative symplectic cohomology, a tool recently introduced by U. Varolgunes, and demonstrate how they can be used to obtain new symplectic rigidity results. Based on joint work with A. Dickstein, Y. Ganor, and L. Polterovich.