Let μ be a probability measure on the set R{double-struck} of real numbers and μ̂(t):=∫R{double-struck}e-itλdμ(λ) (tεR{double-struck}) be the Fourier transform of μ (i is the imaginary unit). Then, under suitable conditions, asymptotic formulae for |μ̂(t/x)|2x 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→∞. © 2013 Elsevier Ltd.