A positive Borel measure <$Emu> on <$Eroman bold R> which possesses all power moments, is N-extremal if the space of all polynomials is dense in <$EL sup 2 ( mu )>. If, in addition, <$Emu> generates an indeterminate Hamburger moment problem, then it is discrete. It is known that the class of N-extremal measures that generate an indeterminate moment problem is preserved when a finite number of mass points are moved (not "removed"!). We show that this class is preserved even under change of infinitely many mass points if the perturbations are asymptotically small. Thereby "asymptotically small" is understood relative to the distribution of sup <$Emu>; for example, if supp <$Emu~=~{n sup sigma log~n~:~n~symbol <174>~roman bold N>} with some <$Esigma~>>~2>, then shifts of mass points behaving asymptotically like, e.g. <$En sup { sigma -2 } [log~log~n] sup -2> are permitted.

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