For an N-extremal solution \(\mu \) to an indeterminate moment problem it is known by a theorem of M. Riesz that the measure \((1+x^2)^{-1}d\mu (x)\) is determinate. For \(0<\alpha <1\) we show by contradiction that there exist indeterminate N-extremal solutions \(\mu \) such that \((1+x^2)^{-\alpha }d\mu (x)\) is determinate, and there exist also indeterminate N-extremal solutions \(\mu \) such that \((1+x^2)^{-\alpha }d\mu (x)\) is indeterminate. Explicit examples of such measures are so far only known when \(\alpha =1/2\) . For indeterminate Stieltjes moment problems and for N-extremal solutions \(\mu \) , we show that \((1+x^2)^{-1/2}d\mu (x)\) is indeterminate except when \(\mu =\mu _F\) is the Friedrichs solution in case of which \((1+x^2)^{-1/2}d\mu _F(x)\) is determinate. We identify the Friedrichs and Krein solutions for some indeterminate Stieltjes moment problems.
