  This paper studies Moore's measurable cohomology theory for locally compact
groups and Polish modules. An elementary dimension-shifting argument is used to
show that all classes in that theory have representatives with considerable
extra topological structure beyond measurability. Using this, for certain
target modules one can also construct a direct comparison map with a different
cohomology theory for topological groups defined by Segal, and show that this
map is an isomorphism.
