  Let $\Sigma$ be a finite alphabet, $\Omega=\Sigma^{\mathbb{Z}^{d}}$ equipped
with the shift action, and $\mathcal{I}$ the simplex of shift-invariant
measures on $\Omega$. We study the relation between the restriction
$\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes
$\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant"
measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of
the convex set $\mathcal{I}_n$ which turns out to be strikingly different when
$d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which
are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic
shifts of finite type, whose unique invariant measure gives rise to extreme
points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also
extreme points which arise in other ways. We show that
$\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they
differ for $n$ large enough. We also show that while in dimension one
$\mathcal{I}_n$ are polytopes with rational extreme points, in higher
dimensions every computable convex set occurs as a rational image of a face of
$\mathcal{I}_n$ for all large enough $n$.
