A projection-valued measure on a measurable space 
,
in the normalized convention, is a function 
from 
to the orthogonal projections
on a complex Hilbert
space 
with 
,
,
and countable additivity in the strong
operator topology (Williams 2018, pp. 15-16). Here 
is the identity operator,
and each 
is a bounded operator satisfying 
, where the star denotes the adjoint
operator. Explicitly, for pairwise disjoint measurable sets 
,
 |
for every vector 
, with convergence in the norm
of 
.
Convergence in operator norm is not required. The
defining properties imply 
for all 
.
In particular, disjoint sets give orthogonal
projections whose product is zero (Williams 2018, p. 16).
For 
,
the scalar set function 
is a complex
measure. When 
, it is a positive measure
of total mass 
. Consequently, a unit vector
gives a probability measure (Williams 2018,
p. 16).
The spectral theorem associates to each bounded Hermitian operator 
a unique projection-valued measure on the Borel
sets of its operator spectrum 
such that
 |
This is an operator-valued integral, characterized by 
for
all 
(Williams 2018, pp. 17-20).
