  We study Borel systems and continuous systems of measures, with a focus on
mapping properties: compositions, liftings, fibred products and disintegration.
Parts of the theory we develop can be derived from known work in the
literature, and in that sense this paper is of expository nature. However, we
put the above notions in the spotlight and provide a self-contained, purely
measure-theoretic, detailed and thorough investigation of their properties, and
in that aspect our paper enhances and complements the existing literature. Our
work constitutes part of the necessary theoretical framework for categorical
constructions involving measured and topological groupoids with Haar systems, a
line of research we pursue in separate papers.
