  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
$\sigma$-algebra of their topology. The main examples of MTspaces are
measurable foliations. This is a singular version of the measurable simplicial
cohomology defined by Heitsch and Lazarov for foliations and extended by
Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and
applications to the theory of foliations are given. Moreover we introduce a new
notion of singular L2-cohomology for MT-spaces.
