  Working in Zermelo-Fraenkel Set Theory with Atoms over an
$\omega$-categorical $\omega$-stable structure, we show how \emph{infinite}
constructions over definable sets can be encoded as \emph{finite} constructions
over the Stone-\v{C}ech compactification of the sets. In particular, we show
that for a definable set $X$ with its Stone-\v{C}ech compactification
$\overline{X}$ the following holds: a) the powerset $\mathcal{P}(X)$ of $X$ is
isomorphic to the finite-powerset $\mathcal{P}_{\textit{fin}}(\overline{X})$ of
$\overline{X}$, b) the vector space $\mathcal{K}^X$ over a field $\mathcal{K}$
is the free vector space $F_{\mathcal{K}}(\overline{X})$ on $\overline{X}$ over
$\mathcal{K}$, c) every measure on $X$ is tantamount to a \emph{discrete}
measure on $\overline{X}$. Moreover, we prove that the Stone-\v{C}ech
compactification of a definable set is still definable, which allows us to
obtain some results about equivalence of certain formalizations of register
machines.
