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(''X'', ''M'') is a Mapping π from ''M'' to the set of Self-adjoint Projections on a Hilbert Space ''H'' such that

: \pi(X) = \operatorname{id}_H \quad

and for every ξ, η ∈ ''H'', the set-function

:\operatorname{S}_\pi(\xi, \eta)(A) = \langle \pi(A)\xi \mid \eta angle

is a Complex Measure on ''M'' (that is, a complex-valued Countably Additive function).

If π is a projection-valued measure and

: A \cap B = \emptyset,

then π(''A''), π(''B'') are Orthogonal Projection s. From this follows that in general,

: \pi(A) \pi(B) = \pi(A \cap B).

Example. Suppose (''X'', ''M'', μ) is a measure space. Let π(''A'') be the operator of multiplication by the Indicator Function 1''A'' on ''L''2(''X''). π is a projection-valued measure.


EXTENSIONS OF PROJECTION-VALUED MEASURES


If π is an additive projection-valued measure on (''X'', ''M''), then the map

: \mathbf{1}_A \mapsto \pi(A)

extends to a linear map on the vector space of step functions on ''X''. In fact, it is easy to check that this map is a ring homomorphism. In fact this map extends in a canonical way to all bounded complex-valued Borel functions on ''X''.

Theorem. For any bounded ''M''-measurable function ''f'' on ''X'', there is a unique bounded linear operator Tπ(''f'') such that

: \langle \operatorname{T}_\pi(f) \xi \mid \eta angle = \int_X f(x) d \operatorname{S}_\pi (\xi,\eta)(x)

for all ξ, η ∈ ''H''. The map

: f \mapsto \operatorname{T}_\pi(f)

is a homomorphism of rings.


STRUCTURE OF PROJECTION-VALUED MEASURES


First we provide a general example of projection-valued measure based on Direct Integral s. Suppose (''X'', ''M'', μ) is a measure space and let {''H''''x''}''x'' ∈ ''X'' be a μ-measurable family of Hilbert spaces. For every ''A'' ∈ ''M'', let π(''A'') be the operator of multiplication by 1''A'' on the Hilbert space

: \int_X^\oplus H_x \ d \mu(x).

Then π is a projection-valued measure on (''X'', ''M'').

Suppose π, ρ are projection-valued measures on (''X'', ''M'') with values in the projections of ''H'', ''K''. π, ρ are unitarily equivalent Iff there is a unitary operator ''U'':''H'' → ''K'' such that

  • ho(A) U \quad


for every ''A'' ∈ ''M''.

Theorem. If (''X'', ''M'') is a Standard Borel Space , then for every projection-valued measure π on (''X'', ''M''), there is a Borel measure μ and a μ-measurable family of Hilbert spaces {''H''''x''}''x'' ∈ ''X'' , such that π is unitarily equivalent to multiplication by 1''A'' on the Hilbert space

: \int_X^\oplus H_x \ d \mu(x).

The measure class of μ and the measure equivalence class of the multiplicity function ''x'' → dim ''H''''x'' completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure π is ''homogeneous of multiplicity'' ''n'' iff the multiplicity function has constant value ''n''. Clearly,

Theorem. Any projection-valued measure π is an orthogonal direct sum of homogeneous projection-valued measures:

  :<math> H N \int_{X_n}^\oplus H_x \ d (\mu X_n) (x) </math>