  We investigate the following questions: Given a measure $\mu_\Lambda$ on
configurations on a subset $\Lambda$ of a lattice $\mathbb{L}$, where a
configuration is an element of $\Omega^\Lambda$ for some fixed set $\Omega$,
does there exist a measure $\mu$ on configurations on all of $\mathbb{L}$,
invariant under some specified symmetry group of $\mathbb{L}$, such that
$\mu_\Lambda$ is its marginal on configurations on $\Lambda$? When the answer
is yes, what are the properties, e.g., the entropies, of such measures? Our
primary focus is the case in which $\mathbb{L}=\mathbb{Z}^d$ and the symmetries
are the translations. For the case in which $\Lambda$ is an interval in
$\mathbb{Z}$ we give a simple necessary and sufficient condition, local
translation invariance (LTI), for extendibility. For LTI measures we construct
extensions having maximal entropy, which we show are Gibbs measures; this
construction extends to the case in which $\mathbb{L}$ is the Bethe lattice. On
$\mathbb{Z}$ we also consider extensions supported on periodic configurations,
which are analyzed using de~Bruijn graphs and which include the extensions with
minimal entropy. When $\Lambda\subset\mathbb{Z}$ is not an interval, or when
$\Lambda\subset\mathbb{Z}^d$ with $d>1$, the LTI condition is necessary but not
sufficient for extendibility. For $\mathbb{Z}^d$ with $d>1$, extendibility is
in some sense undecidable.
