nLab 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

  • all operators in the image are selfadjoint P(A)=P(A) *P(A) = P(A)^*

  • P(A 1A 2)=P(A 1)P(A 2)P(A_1\cap A_2) = P(A_1) P(A_2) for all A 1,A 2BA_1,A_2\in B where the product is the composition of the operators

  • P(A 1A 2)=P(A 1)+P(A 2)P(A_1\cup A_2) = P(A_1)+P(A_2) for all A 1,A 2BA_1,A_2\in B such that A 1A 2=A_1\cap A_2 = \emptyset

  • if A nAA_n\to A, in the sense of coinciding upper and lower limit of sets, A= n knA k= n knA kA= \cap_n \cup_{k\geq n} A_k = \cup_n \cap_{k\geq n} A_k, then P(A n)P(A)P(A_n)\to P(A) in the strong operator topology. (note: check if strong)

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.


  • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]

  • A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

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