Extensions of Projection-valued Measures
If π is an additive projection-valued measure on (X, M), then the map
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
for all ξ, η ∈ H. The map
is a homomorphism of rings.
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