A short proof of ergodicity of Babillot-Ledrappier measures

Let $M$ be a compact manifold, and let ${\phi_t}$ be a transitive homologically full Anosov flow on $M$. Let $\widetilde{M}$ be a $\mathbb{Z}^d$-cover for $M$, and let $\widetilde{\phi_t}$ be the lift of ${\phi_t}$ to $\widetilde{M}$. Babillot and Ledrappier exhibited a family of measures on $\widetilde{M}$, which are invariant and ergodic with respect to the strong stable foliation of $\widetilde{\phi_t}$. We provide a new short proof of ergodicity.