CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

A setisshyorHaar null(in the sense of Christensen) if there exists a Borel setand a Borel probability measureμonC[0, 1] such thatandfor allf∈C[0, 1]. The complement of a shy set is called aprevalentset. We say that a set isHaar ambivalentif it is neither shy nor prevalent.The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy manyf∈C[0, 1]?The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category)f∈C[0, 1] from the topological point of view. We prove that the functionsf∈C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balkaet al. proved that the functionsf∈C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set inC[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functionsf∈C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦f–1are not perfect form a Haar ambivalent set inC[0, 1].We show that for the prevalentf∈C[0, 1] for the genericy∈f([0, 1]) the level setf–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functionsf∈C[0, 1] for which there exists a perfect setPf⊂ [0, 1] such thatfʹ(x) = ∞ for allx∈Pfis Haar ambivalent.

Haar Null Sets and the Consistent Reflection of Non-meagreness

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.