The Nevalinna transform Ka,ρ(z) of a positive measure ρ and a constant a, plays an important role in complex analysis and – more recently – in the context of the boolean convolution. We show here that its restriction to the imaginary axis, ka,ρ(it), can be expressed as the Laplace transform of the Fourier transform (a characteristic function) of ρ. Consequently, ka ρ is sufficient for the unique identification of the measure ρ and the constant a. Finally, we identify a relation between the free additive Voiculescu ⊞ and boolean ⊎ convolutions.
Random integral mappings give isomorphism between the subsemigroups of the classical (I D, *) and the free-infinite divisible (I D, ⊞) probability measures. This allows us to introduce new examples of such measures, more precisely their corresponding characteristic functionals.