  This paper is concerned with the theoretical understanding of $\alpha$-stable
sheets $U$ on $\mathbb{R}^d$. Our motivation for this is in the context of
Bayesian inverse problems, where we consider these processes as prior
distributions, aiming to quantify information of the posterior. We derive
convergence results referring to finite-dimensional approximations of
infinite-dimensional random variables. In doing so we use a number of variants
which these sheets can take, such as a stochastic integral representation, but
also random series expansions through Poisson processes. Our proofs will rely
on the fact of whether $U$ can omit $L^p$-sample paths. To aid with the
convergence of the finite approximations we provide a natural discretization to
represent the prior. Aside from convergence of these stable sheets we address
whether both well-posedness and well-definiteness of the inverse problem can be
attained.
