On Zero-one Laws

An extension of a topological zero-one law due to M. and K.P.S. Bhaskara Rao and of the Hewitt-Savage zero-one law is presented. In [8] M. and K.P.S. Bhaskara Rao proved the following topological zeroone law, which we shall refer to as the Bhaskara Rao zero-one law. Theorem 1. Let X be a topological space and let $ be a group of homeomorphisms of X onto itself with the property that for any two nonempty open sets U and V, there exists a homeomorphism <p £ $ such that <p(t/) n V # 0. // S C X is a set with the classical Baire property such that <p(S ) = S for all (¡p £ $, then either S or X — S is of the first category. This theorem was first stated by Oxtoby [7] for a complete separable metric space without isolated points and a cyclic group; a proof in the case of a complete metric space and any countable group was given in [2, p. 75]. A theorem very closely related to that of [8] was also found, independently of M. and K.P.S. Bhaskara Rao, by Kuratowski (see [5, Theorem 1]); in fact by a slight modification of the proof of the latter theorem, one easily obtains the implication (1)-» (4) of [8]. Using Theorem 1, M. and K.P.S. Bhaskara Rao obtained a category analogue of the Hewitt-Savage zero-one law. The analogy here is, however, incomplete, since the measure analogue of their category analogue is not the Hewitt-Savage zero-one law itself, but an extension of this theorem to symmetric sets measurable with respect to the completion of a product probability space. In this paper an abstract zero-one law is established which provides a further extension of the Bhaskara Rao zero-one law and also furnishes the proper measure-theoretic analogue. 1. Terminology. It has been shown in [6] that several analogies between category and measure can be unified within an abstract theory of Baire category starting with the notion of a ^-family. In this section we recall some basic definitions and facts from [6] which are pertinent to this paper. Received by the editors February 18, 1976 and, in revised form, July 7, 1976. AMS (MOS) subject classifications (1970). Primary 54A05, 60F20; Secondary 28A05.