IFS measures on generalized Bratteli diagrams

The paper’s Theorem 3.4 establishes exactly the equivalence the model proves: a Borel probability measure ν on the path space XB satisfies the IFS identity ν = ∑e pe(ν ∘ τe−1) if and only if the vertex-probability vector q with entries qv = ν([v]) solves Pq = q, where P has entries ps(e),r(e) (edges identified with ordered pairs) . The paper constructs ν on cylinders via ν([f0,…,fn−1]) = pf0⋯pfn−1 qr(fn−1), checks Kolmogorov consistency, extends to Borel sets, and verifies the IFS identity on cylinders (hence on the σ-algebra) . The model’s solution follows the same structure (cylinders → Kolmogorov extension → equality on cylinders → π–λ), with slightly stronger assumptions (it explicitly requires pe>0 for all e and emphasizes full support). Notationally and logically, the two arguments coincide in substance and scope.