  We adapt Gromov's notion of ideal-valued measures to symplectic topology, and
use it for proving new results on symplectic rigidity and symplectic
intersections. Furthermore, it enables us to discuss three "big fiber
theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber
Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic
topology, from a unified viewpoint. Our main technical tool is an enhancement
of the symplectic cohomology theory recently developed by Varolgunes.
