Projection (or: projection-valued) measures are operator-valued measures of a special type. They appear for example in the theory of reproducing kernel Hilbert spaces, coherent states and the foundations of quantum mechanics. A projection measure is used to parametrize a complete family of projection operators by subsets of some parameter space.
Given a set and some -algebra of subsets of , with , and a complex Hilbert space , a map is called a projection-(valued) measure on with values in if
all operators in the image are selfadjoint
for all where the product is the composition of the operators
for all such that
if , in the sense of coinciding upper and lower limit of sets, , then in the strong operator topology. (note: check if strong)