INVERSE LIMIT METHOD FOR GENERALIZED BRATTELI DIAGRAMS AND INVARIANT MEASURES

The paper proves that for the triangular generalized Bratteli diagram B∞ the ergodic tail-invariant probability measures are exactly the one-parameter family {μ_a : a>0}, with cylinder/tower weights q^{(n)}_{a,i} = C(i+n−2, n−1) (a^{i−1}/(a+1)^{n+i−1}); this is established via an inverse-limit framework and a Hausdorff moment/complete monotonicity argument (Proposition 8.3 and Theorem 8.7) . The candidate solution reaches the same classification but by a different route: exchangeability/de Finetti for the increments and a functional-equation forcing a geometric law, yielding the same μ_a and negative-binomial tower weights. The only caveat is a minor gap when the model implicitly imposes the “equal-weight-for-fixed-sum” constraint at the component (i.i.d.) level in the de Finetti mixture; this is not logically necessary for non-ergodic measures, although the conclusion that every tail-invariant measure is a mixture of μ_a still follows from standard ergodic decomposition for tail equivalence relations and the paper’s classification of ergodic points. Overall, both are correct and consistent, using different proof methods.