Approximation and invariant measures

Abstract

We prove certain approximation theorems for the class of invertible, measurable, and non-singular transformations of the unit interval. The main results concern the approximation of such transformations by those having no a-finite invariant measure absolutely continuous with respect to Lebesgue measure. We are indebted to A. ION~SCTJ TULC~A for making available to us a preprint of her paper [6] which was helpful to us. The major differences and points of contact between the results of [6] and those of the present paper are discussed below. Theorem i of the present paper is a generalization of a theorem of V. A. l~o~LI~ (see [2], p. 75). Theorem 1 states that every antiperiodic transformation can be approximated by a strictly periodic transformation in the uniform topology, defined by the metric d (~, a) = ~n (T # cr), where the degree of approximation depends on the period of the approximating transformation. X~OtILIN's theorem is for the special case where the antiperiodie transformation is measure preserving whereas in fact theorem 1 includes the case where the antiperiodic transformation may not have a a-finite invariant measure absolutely continuous with respect to Lebesgue measure. Our proof is essentially different from the proof for the measure preserving case in [2]. As a corollary of theorem 1 we obtain an unpublished result of C. E. LINDERHOLM which is stated without proof in [6]. Every invertible, measurable, and non-singular transformation T of the unit interval defines a transformation ~T of the essentially bounded measurable functions characterized by ~T~0~ = ~v~-l(~) where ~A is the characteristic function of the set A. Furthermore the adjoint transformation T~ is a positive invertible isometry of L1 [0,1]. It can be readily obtained from the Bm~orF ergodic theorem that r has no a-finite invariant measure absolutely continuous with respect to Lebesgue measure if the limit �9 1 n-1