Integration of Laplace transform

Question

Is it possible to evaluate/simplify the following integral involving the Laplace transform? If yes, what are the steps or hints? $$ I = \int_{0}^{\infty}\left[\,1 - \mathcal{L}_{X}\left(a \over y\right)\right]^{k} y^{b-1}\,\mathrm{d}y, $$ where $\mathcal{L}_{X}\left(\cdot\right)$ is the Laplace transform of a positive random variable $X$. $\quad a > 0,\ b \in \left[0, 1\right],\ $ and $\ k \in \mathbb{N}$.

Answer 1

The Laplace transform of $\dfrac{1}{y}$ fails to exist. Look here for the details