Translation invariant extensions of finite volume measures

We investigate the following questions: Given a measure $\mu_\Lambda$ on configurations on a set $\Lambda\subset \mathbb{Z}^d$, where a configuration is an element of $\Omega^\Lambda$ for some fixed set $\Omega$, does there exist a translation invariant measure $\mu$ on configurations on all of $\mathbb{Z}^d$ such that $\mu_\Lambda$ is its projection on $\Lambda$? When the answer is yes, what are the properties, e.g., the entropies, of such measures? For the case in which $\Lambda$ is an interval in $\mathbb{Z}$ we give a simple necessary and sufficient condition, pre-translation-invariance (PTI), for extendibility. For PTI measures we construct extensions having maximal entropy, which we show are Gibbs measures. 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 PTI condition is necessary but not sufficient for extendibility; we give examples but leave most questions open.