  The axiomatic theory of quantum first-kind measurements is developed in a
rigorous form based on five Postulates. The measurement theory for observable
with continuous spectrum is given in a rigged Hilbert space. This approach also
describes the measurements with non-ideal initial conditions. It yields the
survival effect in the position measurement of the particles. It is also found
that no such survival effect in the momentum measurement of the particles. The
Postulates of axiomatic theory yield the survival effect which violates the
Heisenberg uncertainty relation. This theoretical result is demonstrated by the
wave function with minimum of position and momentum uncertainty of the
particle.
