On the moments of distributions attracted to stable laws

The following theorem is proved. Let the distribution function $F(x)$ belong to the domain of the normal attraction of a stable law with exponent $\alpha$, $0<\alpha<2$. If $\delta>0$ and $\psi(x)$ is an even function which is positive and nondecreasing on the half-line $x\ge\delta$, then convergence of the integral $\int_\delta^\infty\frac{dx}{x\psi(x)}$ is equivalent to convergence of the integral $\int_{|x|\ge\delta}\frac{|x|^\alpha\,dF(x)}{\psi(x)}$.