The absolute of the comb graph

In the 1970s R. Stanley introduced the comb graph $\mathbb{E}$ whose vertices are indexed by the set of compositions of positive integers and branching reflects the ordering of compositions by inclusion. A. Vershik defined the absolute of a $\mathbb{Z}_+$-graded graph as the set of all ergodic probability central measures on it. We show that the absolute of $\mathbb{E}$ is naturally parametrized by the space $\Omega = \{(\alpha_1, \alpha_2, \dots ) : \alpha_i \ge 0$, $\sum_i \alpha_i \le 1\}$.