Cohomology of measurable laminations

A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel σ-algebra of their topology. The main examples of MT-spaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations (Heitsch and Lazarov, 1991, [7]) and extended by Bermúdez for MT-spaces (Bermúdez, 2006, [1]). Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L 2 -cohomology for MT-spaces.