Finitely additive measures on the invariant foliations of Anosov diffeomorphisms

Consider a $C^3$-smooth orientation preserving Anosov diffeomorphism on a compact Riemannian manifold.
We shall consider finitely additive measures defined on Borel subsets of the unstable foliation with piecewise $C^1$ boundary. The finitely additive measures on the invariant foliations of dynamical systems were first studied in the series of works of Bufetov on qualitative asymptotics of the ergodic integrals along the orbits of translation flows on flat surfaces. The method was extended to the setting of horocycle flows on compact Riemann surfaces of constant negative curvature [Bufetov-Forni] and of translation flows occurring in tiling dynamics [Bufetov-Solomyak].
In our case, the finitely additive measures are used for establishing the following qualitative equidistribution theorem for the unstable leaves: for any $C^2$ function with zero average with respect to the measure of maximal entropy, its integrals over the iterations of a unit unstable ball, after a proper normalization, uniformly converge to a finitely additive measure of the ball. This theorem extends the classical results on equidistribution of the unstable leaves due to Margulis.
We use the approach of Gouezel and Liverani to construct such the Banach space of currents inducing the regular finitely additive measures. Analyzing the spectrum of the transfer operator acting on this Banach space, we obtain the asymptotic expansion for the leafwise integrals.
Comparing to the Banach spaces constructed in the works of Baladi and Tsujii, who considered the rate of decay/growth of the Fourier transform of distributions in the stable/unstable cones, and the ones constructed by Faure and Sjoestrand by tools of semiclassical analysis, the spaces of Gouezel and Liverani inherit the leafwise structure of the distributions, and allow us to consider the property of invariance under the holonomy along the stable leaves. We prove that the eigenfunctions of the transfer operator with the eigenvalues close to the spectral radius induce finitely additive measures on the unstable leaves, invariant under stable holonomy. The holonomy invariant finitely additive measures control the leafwise integrals.