Formal Definition
A projection-valued measure on a measurable space (X, M), where M is a σ-algebra of subsets of X, is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that
and for every ξ, η ∈ H, the set-function
is a complex measure on M (that is, a complex-valued countably additive function). We denote this measure by .
If π is a projection-valued measure and
then π(A), π(B) are orthogonal projections. From this follows that in general,
Example. Suppose (X, M, μ) is a measure space. Let π(A) be the operator of multiplication by the indicator function 1A on L2(X). Then π is a projection-valued measure.
Read more about this topic: Projection-valued Measure
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