  We show the compatibility of the differential geometric and the topological
construction of equivariant characteristic classes for compact Lie groups. Our
analysis motivates a differential geometric construction for equivariant
characteristic classes in the non-compact case. This compatibility is generally
assumed and used in various cases, but there is only a proof for compact
connected Lie groups in the literature, see the work of Raoul Bott and Loring
Tu. Our proof applies and generalizes ideas of Johan Dupont and Ezra Getzler.
