  There are many research works devoted to Gibbs measure for models on Cayley
trees. Among these works, there are some works in which the general results are
identical, but the considered models are various. In this article, we present
the construction of Gibbs measures in the language of measure theory and reply
to the question ``When can we construct Gibbs specifications?" Also, we present
a new condition (convenient for verification) which is equivalent to the
consistency condition of kernels. The obtained results in the article are
general, not for a particular model. On the contrary, these results hold for
some considered models on Cayley trees.
