  We consider non-negative $\sigma$-finite measure spaces coupled with a proper
functional $P$ that plays the role of a perimeter. We introduce the Cheeger
problem in this framework and extend many classical results on the Cheeger
constant and on Cheeger sets to this setting, requiring minimal assumptions on
the pair measure space-perimeter. Throughout the paper, the measure space will
never be asked to be metric, at most topological, and this requires the
introduction of a suitable notion of Sobolev spaces, induced by the coarea
formula with the given perimeter.
