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

Let μ be a probability measure on the set R of real numbers and μ ˆ ( t ) : = ∫ R e − i t λ d μ ( λ ) ( t ∈ R ) be the Fourier transform of μ ( i is the imaginary unit). Then, under suitable conditions, asymptotic formulae for | μ ˆ ( t / x ) | 2 x in 1 / x as 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 N of measurements made in a time interval as N → ∞ .