In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let f be an S-unimodal map of the interval having no limit cycles. Then f is ergodic with repect to the Lebesque measure, and has a unique attractor A in the sense of Milnor. This attractor coincides with the conservative kernel of f. There are no strongly wandering sets of positive measure. If f has a finite a.c.i. (absolutely continuous invariant) measure u, then it has positive entropy. This result is closely related to the following: the measure of Feigenbaum-like attractors is equal to zero. Some extra topological properties of Cantor attractors are studied.