  We construct the covariant nonlocal action for recently suggested
long-distance modifications of gravity theory motivated by the cosmological
constant and cosmological acceleration problems. This construction is based on
the special nonlocal form of the Einstein-Hilbert action explicitly revealing
the fact that this action within the covariant curvature expansion begins with
curvature-squared terms. Nonlocal form factors in the action of both quantum
and brane-induced nature are briefly discussed. In particular, it is emphasized
that for certain class of quantum initial value problems nonlocal nature of the
Euclidean action does not contradict the causality of effective equations of
motion.
