  Let $G$ be a group and assume that $(A_p)_{p\in G}$ is a family of algebras
with identity. We have a {\it Hopf $G$-coalgebra} (in the sense of Turaev) if,
for each pair $p,q\in G$, there is given a unital homomorphism
$\co_{p,q}:A_{pq}\to A_p \ot A_q$ satisfying certain properties.
  Consider now the direct sum $A$ of these algebras. It is an algebra, without
identity, except when $G$ is a finite group, but the product is non-degenerate.
The maps $\co_{p,q}$ can be used to define a coproduct $\co$ on $A$ and the
conditions imposed on these maps give that $(A,\co)$ is a multiplier Hopf
algebra. It is $G$-cograded as explained in this paper.
  We study these so-called {\it group-cograded multiplier Hopf algebras}. They
are, as explained above, more general than the Hopf group-coalgebras as
introduced by Turaev. Moreover, our point of view makes it possible to use
results and techniques from the theory of multiplier Hopf algebras in the study
of Hopf group-coalgebras (and generalizations).
  In a separate paper, we treat the quantum double in this context and we
recover, in a simple and natural way (and generalize) results obtained by
Zunino. In this paper, we study integrals, in general and in the case where the
components are finite-dimensional. Using these ideas, we obtain most of the
results of Virelizier on this subject and consider them in the framework of
multiplier Hopf algebras.
