Average measure theoretic entropy for a family of expanding on average random Blaschke products

The paper proves Theorem 1.3 by (i) expressing the fibre entropy with respect to the random acim as an integral of log|T′| via general results (Corollary 3.4, building on Buzzi’s framework), and (ii) showing that the θ-average of the invariant densities equals Lebesgue using Lemma 4.1 and Proposition 4.2; this yields the stated formula for the θ-averaged entropy h̄(T) . The candidate solution proves the same result with a different route: Step 1 uses a fibrewise Rokhlin/Jacobian identity to obtain hfib(Tθ)=∫∫log|T′| dμθ; Step 2 shows that the θ-average of the Poisson kernels P_{x_{ω,θ}} is Lebesgue by a Fourier-moment vanishing argument; Step 3 integrates and adds base entropy. This matches the paper’s conclusion. Minor caveat: the model asserts the θ-averaging identity for every ω, whereas the paper establishes it for P-a.e. ω (which suffices) .