Let \(\mu\) be a positive measure on the real axis. A measure \(\mu\) is determinate if no other measure has the same moments as those of \(\mu\), otherwise \(\mu\) is indeterminate. Let \(M_0\) denote the set of measures having a finite number of real points as support. A measure \(\mu\) has an infinite index of determinacy if for any measure \(\mu_0\in M_0\), the measure \(\mu+ \mu_0\) is determinate. The following question was posed by Christian Berg [cf. \textit{C. Berg} and \textit{A. J. Duran}, Proc. Am. Math. Soc. 125, No. 2, 523-530 (1997; Zbl 0889.47010)]. Suppose the measure \(\mu\) has infinite index of determinacy and the measure \(\nu\) has a compact support. Is it true that the measure \(\mu+ \nu\) is indeterminate? In the given note a positive answer to this question is given. Moreover, the statement is fulfilled if the Fourier transform of the measure \(\nu\) is analytic in some strip around the real axis. The last proposition was formulated, as a conjecture, by \textit{M. Sodin} [J. Anal. Math. 69, 293-305 (1996; Zbl 0867.41009)].