 Second, for each fixed normalized vector state formula_51, the association 

is a probability measure on "X" making the values of the observable into a random variable.A measurement that can be performed by a projection-valued measure π is called a projective measurement.If "X" is the real number line, there exists, associated to π, a Hermitian operator "A" defined on "H" by

which takes the more readable form

if the support of π is a discrete subset of R. 

The above operator A is called the observable associated with the spectral measure.