Maxim R. Burke

SETS ON WHICH MEASURABLE FUNCTIONS ARE DETERMINED BY THEIR RANGE

We study sets on which measurable real-valued functions on a measur- able space with negligibles are determined by their range. 1. Introduction. In (BD, Theorem 8.5), it is shown that, under the Continuum Hypothesis (CH), in any separable Baire topological space X there is a set M such that for any two continuous real-valued functions f and g on X ,i f fand g are not constant on any nonvoid open set then f (M) g(M) implies f = g. In particular, if f (M )= g ( M ) then f = g. Sets having this last property with respect to entire (analytic) functions in the complex plane were studied in (DPR) where they were called sets of range uniqueness (SRUs). We study the properties of such sets in measurable spaces with negligibles. (See below for the definition.) We prove a generalization of the aforementioned result (BD, Theorem 8.5) to such spaces (Theorem 4.3) and answer Question 1 from (BD) by showing that CH cannot be omitted from the hypothesis of their theorem (Example 5.17). We also study the descriptive nature of SRUs for the nowhere constant continuous functions on Baire Tychonoff topological spaces. When X = R, the result of (BD, Theorem 8.5) states that, under CH, there is a set M R such that for any two nowhere constant continuous functions f g: R R ,i f f ( M ) g ( M )t hen f = g. It is shown in (BD, Theorem 8.1) that there is (in ZFC) a set M R such that for any continuous functions f g: R R ,i f fhas countable level sets and g(M) f (M )t hen gis constant on the connected components of x R : f (x) = g(x) . In the case where g is the identity function, these properties of M are similar to various properties that have been considered in the literature. Dushnik and Miller (DM) showed that, under CH, there is an uncountable set M R such that for any monotone (nonincreasing or nondecreasing) function f : R R ,i f x R : f ( x )= x is nowhere dense, then f (M) M is countable. Building on this result from (DM), B ¨ uchi (B¨

Sets of range uniqueness for classes of continuous functions

Diamond, Pomerance and Rubel (1981) proved that there are subsets M of the complex plane such that for any two entire functions f and g if f [M ] = g[M ], then f = g. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set M ⊂ R for the class Cn(R) of continuous nowhere constant functions from R to R, while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of C(R), including the class D1 of differentiable functions and the class AC of absolutely continuous functions, a set M with the above property can be constructed in ZFC. We will also prove the existence of a set M ⊂ R with the dual property that for any f, g ∈ Cn(R) if f−1[M ] = g−1[M ], then f = g.

A proof of Hechler's theorem on embedding \(\aleph_1\)-directed sets cofinally into \((\omega^\omega,<^*)\)

Abstract. We give a proof of Hechlers theorem that any

Continuous functions which take a somewhere dense set of values on every open set

\aleph_1

Entire functions mapping uncountable dense sets of reals onto each other monotonically

-directed partial order can be embedded via a ccc forcing notion cofinally into

Characterizing uniform continuity with closure operations

\omega^\omega

Hechler's theorem for the null ideal

ordered by eventual dominance. The proof relies on the standard forcing relation rather than the variant introduced by Hechler.

Linear liftings for non-complete probability spaces

Abstract We study the class of Tychonoff spaces that can be mapped continuously into R in such a way that the preimage of every nowhere dense set is nowhere dense. We show that every metric space without isolated points is in this class. We also give examples of spaces which have nowhere constant continuous maps into R and are not in this class.

Liftings for Noncomplete Probability Spacesa

When A and B are countable dense subsets of R, it is a well-known result of Cantor that A and B are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R of an entire function. J.E. Baumgartner showed that consistently 2 N 0 > N 1 and any two subsets of R having N 1 points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartners result for second category sets was established by S. Shelah. He showed that it is consistent that 2 N 0 > N 1 and second category sets of cardinality N 1 exist while any two sets of cardinality N 1 which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelahs theorem can be taken to be the restriction to R of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer n, a nondecreasing surjection g: R → R of class C n and a positive continuous function e: R → R, we may choose the order-isomorphism f so that for all i = 0, 1,..., n and for all x ∈ R, |D i f(x)-D i g(x)|<∈(x).

Approximation by Entire Functions in the Construction of Order-Isomorphisms and Large Cross-Sections

Abstract The purpose of this paper is to determine which metric spaces X and Y are such that the uniformly continuous maps f : X → Y are precisely the continuous maps between ( X , τ 1 ) and ( Y , τ 2 ) for some new topologies τ 1 and τ 2 on X and Y respectively.