It is well known that $\int_{0}^{1} t^{-θ} d t<\infty$ for $θ\in (0,1)$ and $\int_{0}^{1} t^{-θ} d t=\infty$ for $θ\in [1,\infty)$. Since $t$ can be taken as an $α$-stable subordinator with $α=1$, it is natural to ask whether $\int_{0}^{1} t^{-θ} d S_{t}$ has a similar property when $S_{t}$ is an $α$-stable subordinator with $α\in (0,1)$. We show that $θ=\frac 1α$ is the border line such that $\int_{0}^{1} t^{-θ} d S_{t}$ is finite a.s. for $θ\in (0, \frac 1α)$ and blows up a.s. for $θ\in [\frac1α,\infty)$. When $α=1$, our result recovers that of $\int_{0}^{1} t^{-θ} d t$. Moreover, we give a $p$-th moment estimate for the integral when $θ\in (0,\frac 1α)$.