On the Existence of a Mixing Measure

Introduction. An interesting but apparently neglected generalization due to Doob [l], of the ergodic theorem, is the generalization to transformations which are not necessarily one-to-one. In his paper [l], Doob showed that the transformation of the interval [0, l] into itself given by/(x) =nx (modulo 1), where n — 1 is a positive integer, is ergodic under Lebesgue measure and that the law of large numbers is a trivial corollary of this ergodicity. This mapping is not only ergodic, but may be shown to be strongly mixing (which I shall just call mixing). Also, other similar transformations such as f(x) = 2x (0 g x g 1/2), f(x) = 2 2x (1/2 ^ a: g 1) may similarly be shown to be mixing. In a talk at Chicago on April 26, 1952, S. M. Ulam suggested that for sufficiently smooth transformations of a certain type, there might exist measures continuously related to Lebesgue measure under which such transformations are mixing. The main purpose of this paper is to show the validity of this speculation. The statement in the theorem of the uniqueness of FELi which produces an invariant measure does not imply that there are no more invariant measures which are topologically equivalent to Lebesgue measure. In another paper, On conjugacy of some transformations of the interval, I shall show, among other things, that given any function of this type for which there is an invariant measure assigning every open sub-interval a positive measure, there exists an uncountable number of distinct measures, topologically equivalent to Lebesgue measure, under which the function is mixing. Also, there are many more such measures under which the transformation is invariant but not even ergodic. Definition, (a) Let a transformation / be called invariant under a measure u if and only if given a measurable set E, n (E) = p {x/f(x) E E}. (b) Let a transformation / be called ergodic under a measure u if and only if given measurable sets E and F,