Harmonic Analysis on Graphs via Bratteli Diagrams and Path-Space Measures

The paper’s Theorem 5.16 constructs ρ on rectangles via ρ(A×B)=∫_A P(x,B) dν1(x), sets ν2=ρ∘π2^{-1}=ν1P, disintegrates ρ to obtain Q so that dν1(x)P(x,dy)=dν2(y)Q(y,dx), and verifies ν1P=ν2 and ν2Q=ν1; uniqueness follows from disintegration uniqueness and the rectangle construction (see the statement, construction, and identities (5.9), (5.10), and (5.11)–(5.13) in the paper ). The model follows the same blueprint: define ρ on rectangles and extend by Caratheodory, set ν2=ν1P, apply disintegration to get Q, establish the duality identity and the marginal relations, and argue uniqueness modulo ν2-null sets. Minor differences are expository: the model spells out σ-finiteness of ν2 and the normalization Q(y,X1)=1 a.e.; both are consistent with the paper’s framework and with Proposition 5.25’s characterizations of marginals and total masses .