In 2006, A. Gorodetski proved that central fibers of perturbed skew products are Holder continuous with respect to the base point. In the present paper we give an explicit estimate of the Holder exponent mentioned above. Moreover, we extend the Gorodetski theorem from the case when the fiber maps are close to the identity to a much wider class that satisfy the so-called modified dominated splitting condition. In many cases (for example, in the case of skew products over the solenoid or over linear Anosov diffeomorphisms of a torus), the Holder exponent is close to 1. This allows us in a sense to overcome the so-called Fubini nightmare. Namely, we prove that the union of central fibers that are strongly atypical from the point of view of the ergodic theory, has Lebesgue measure zero, despite the lack of absolute continuity of the holonomy map for the central foliation. For that we revisit the Hirsch-Pugh-Shub theory, and estimate the contraction constant of the graph transform map.