  Measure homology is a variation of singular homology designed by Thurston in
his discussion of simplicial volume. Zastrow and Hansen showed independently
that singular homology (with real coefficients) and measure homology coincide
algebraically on the category of CW-complexes. It is the aim of this paper to
prove that this isomorphism is isometric with respect to the l^1-seminorm on
singular homology and the seminorm on measure homology induced by the total
variation. This, in particular, implies that one can calculate the simplicial
volume via measure homology -- as already claimed by Thurston. For example,
measure homology can be used to prove the proportionality principle of
simplicial volume.
