An Isometry Between Measure Homology and Singular Homology

In Thurston’s notes, he gives two different definitions of the Gromov norm (also called simplicial volume) of a manifold and states that they are equal but does not prove it. Gromov proves it in the special case of hyperbolic manifolds as a consequence of his proof that simplicial volume is proportional to volume. We give a proof for all differentiable manifolds. This version corrects a few typos in an earlier version and formally proves the theorem for differentiable manifolds rather than locally finite simplicial complexes (but very few actual changes have been made).