A complex measure on a measurable space 
is a countably
additive set function 
with 
(Hunter 2011, p. 77). The set
of complex measures on 
forms a vector space. Note that this is not the case
for the more common positive measures. Every
complex measure has finite total variation (Rudin
1987, pp. 116-119). The norm 
makes this vector
space a Banach space.
Using the polar representation of 
, a measurable function 
with 
has Lebesgue
integral
 |
where 
is the unit-modulus density in that representation (Rudin 1987, pp. 124-125
and 129).
Complex measures take values in 
. In contrast, a projection-valued
measure takes values in orthogonal projections
on a Hilbert space 
. Its countable
additivity is interpreted in the strong
operator topology, by applying the operators to
each fixed vector, rather than as convergence in operator
norm. The spectral theorem represents a bounded Hermitian
operator 
using such a projection-valued measure
on the Borel sets of its operator
spectrum (Williams 2018, pp. 15-20).
