DISCONTINUITY OF FOURIER TRANSFORMS OF POISSONIAN TYPE COUNTABLY ADDITIVE MEASURES

Abstract

It is proved in the paper that different natural Fourier transforms (FTs) of the measure mentioned in the title are not continuous with respect to such sufficient topologies as Sazonov and Gross–Sazonov (introduced by Smolyanov [Gross–Sazonov theorem for alternating cylindrical measures, Vestnik Moskov. Univ. (4) (1983) 4–12]) topologies. The motivation of the result is the fact that the FT of the standard Wiener measure is discontinuous in a known sufficient topologies as stated by Smolyanov and Fomin (see e.g., [Measures on Topological Linear Spaces, Uspekhi Mat. Nauk31(4) (1976) 3–56]).