Given a locally compact topological group , a system of imprimitivity on consists of a
unitary representation on a Hilbert space
a locally compact Hausdorff space with continuous left -action
a regular projection-valued measure where is the Borel -algebra of
for all and .
In the above definition, one can replace the projection-valued measure by a -representation of the -algebra by defining , then
On the other hand, any satisfying this property defines a regular projection-valued measure as above.
Remark: A possible extension is to replace by a measurable space with a measurable left action of .
This concept is important in Mackey machinery and in the applications to the study of coherent states and Berezin quantization.