In this paper, we investigate the Green measure for a class of non-Gaussian processes in $\mathbb{R}^{d}$. These measures are associated with the family of generalized grey Brownian motions $B_{β,α}$, $0<β\le1$, $0<α\le2$. This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability $1$ for $dα>2$ and $1<α\le2$. The Green measure then generalizes those measures of all these classes.