Vitali set

Abstract: We introduce the concept of measuroids, and prove the Uniform Boundedness Principle of Nikodym-Grothendieck, the Vitali-Hahn-Saks theorem and the Nikodym Convergence Theorem for measuroids. The theorem in the title asserts that the setwise limit of a sequence of probability measures is again a probability measure. The present note gives some versions of this classical result and of some related results. Most of the ideas in our proofs are quite standard and can be found in the beautiful treatment [3] by Diestel; see also the lecture notes [1] by Antosik-Swartz. In the sequel, let (Q, X#) denote a fixed measurable space. We begin with a proof of the following "primitive" result. Theorem 1. Let (/,n)1) be a sequence of bounded positive measures on X#, and let :X > -[O, oc] be defined by (*) i/ (A) = lim sup /in (A). n -+oo Then we have either (a) yI(An) -? 0 for each sequence (An)' in X# with An 1 0, or (b) inf, yI(Bn) > 0 for some disjoint sequence (Bn)0 in X. Proof. Suppose that (a) is not the case. Then there exists /1 > 0 and Ap E J((p E N) such that Ap I 0 and (1) V/ (AP)> VP 'd>_ We choose natural numbers n1 Pk1 such that gnk (Apkl) > / . Moreover, each lin is a bounded measure and AP 1 0. Hence link(Ap) -? 0 as p -? oo. So we can find Pk > nk such that Received by the editors October 3, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28A33, 28B05; Secondary 46GI0.

Abstract: The concept of measurability of real-valued functions with re- spect to various classes of measures is introduced and the associated notion of an absolutely nonmeasurable function is investigated. A characterization of such functions is given. Also, it is shown that functions produced by the classical Vitali partition of the real line are measurable with respect to the class of all extensions of the Lebesgue measure on this line.

Abstract: It is shown that some Vitali subsets of the real line R can be measurable with respect to certain translation quasi-invariant measures on R extending the standard Lebesgue measure. On the other han...