Invariant measures for reducible generalized Bratteli diagrams

The paper proves that on the BIO/DIO diagram the ergodic probability tail-invariant measures are exactly the normalizations of the canonical extensions µ̂i with finite total mass (Theorem 4.3), using decomposition XB=⨆i X̂Bi and uniqueness of the extension from each odometer Bi. The candidate solution reaches the same classification via an explicit cylinder-weight recursion p(n)=F^T_n p(n+1), a stabilization-by-column argument, and a one-dimensionality cone argument on each X̂Bi. Aside from a minor technical imprecision about an event-measure factor in Step 4, the candidate’s proof is correct and complements the paper’s approach. Thus both are correct, but they follow different (compatible) routes. See the BIO/DIO construction and Claim 4.1 for the partition, and Theorem 2.7 and Theorem 4.3 for the recursion and classification .