The article studies approximations for stable like jump processes
on fractal sets $F\subset\mathbb{R}^{n}$. Processes on $d$-sets
are approximated by jump processes on the $\varepsilon$-parallel
sets. For the special case of self-similar sets with equal
contraction ratios, approximations in terms of finite Markov
chains are provided. In either case, the convergence of Dirichlet
forms, semigroups and resolvents are established as well as
the convergence of the finite-dimensional distributions under
canonical initial distributions. In the self-similar case
also the weak convergence of the laws under these initial
distributions in $D_{F}([0,t_{0}])$ is proved.