  Measure homology was introduced by Thurston in his notes about the geometry
and topology of 3-manifolds, where it was exploited in the computation of the
simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently
proved that there exists a canonical isomorphism between measure homology and
singular homology (on the category of CW-complexes), and it was then shown by
Loeh that, in the absolute case, such isomorphism is in fact an isometry with
respect to the L^1-seminorm on singular homology and the total variation
seminorm on measure homology. Loeh's result plays a fundamental role in the use
of measure homology as a tool for computing the simplicial volume of Riemannian
manifolds.
  This paper deals with an extension of Loeh's result to the relative case. We
prove that relative singular homology and relative measure homology are
isometrically isomorphic for a wide class of topological pairs. Our results can
be applied for instance in computing the simplicial volume of Riemannian
manifolds with boundary.
  Our arguments are based on new results about continuous (bounded) cohomology
of topological pairs, which are probably of independent interest.
