projection measure

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 XX and some σ\sigma-algebra BB of subsets of XX, with XBX\in B, and a complex Hilbert space HH, a map P:BEndHP: B\to End H is called a projection-(valued) measure on BB with values in EndHEnd H if

Typical example is that (X,τ)(X,\tau) is a topological space and BB is the σ\sigma-algebra (X)\mathcal{B}(X) of Borel subsets of XX.