The problem of infinite information flow

The paper’s Theorem A states that if, for a positive-measure set of vt, the conditional law μv = P(Vt+1 ∈ · | Vt = v) charges an atomless continuum, then the transfer entropy TU→V,t = I(Vt+1; Ut | Vt) is infinite; it reduces the claim to an averaged conditional-MI statement via disintegration (Theorem B/Prop. 2.8), and then applies a per-slice MI divergence result (Theorem 3.6) to conclude infinity (Theorem 3.7) . The candidate’s solution proves, for each v with atomless μv, singularity of the true conditional joint law on the graph Γv relative to the product μv × λv and hence an infinite per-v KL divergence, then integrates over v to conclude I(Vt+1; Ut|Vt)=∞. This is the same graph-based singularity argument used in the paper’s proofs of Theorem 3.6 (two-variable case) and Theorem 3.7 (cMI via disintegration), specialized to TE defined as I(Vt+1; Ut|Vt) . Both are correct and essentially the same proof structure (graph support vs. product reference, absolute-continuity failure, and KL = ∞).