Asymptotic analysis of the Fourier transform of a probability measure with application to the quantum Zeno effect

Let μμ be a probability measure on the set RR of real numbers and μˆ(t):=∫Re−itλdμ(λ) (t∈Rt∈R) be the Fourier transform of μμ (ii is the imaginary unit). Then, under suitable conditions, asymptotic formulae for |μˆ(t/x)|2x in 1/x1/x as x→∞x→∞ are derived. These results are applied to the so-called quantum Zeno effect to establish asymptotic formulae for its occurrence probability in the inverse of the number NN of measurements made in a time interval as N→∞N→∞.