On continuity of measurable group representations and homomorphisms (Q2919694)

The author is concerned with non-separable Hilbert space so chooses to investigate consistency with related axiomatic systems. \textit{K. Gödel} [The consistency of the continuum hypothesis. Princeton, N. J.: Princeton University Press (1940; Zbl 0061.00902)] showed that the continuum hypothesis is consistent with the axioms of set theory. The independence of the continuum hypothesis from ZFC (Zermelo-Fraenkel, axiom of choice) and of the independence of the axiom of choice from the remaining ZFC axioms follows from \textit{K. Gödel} [Am. Math. Mon. 54, 515--525 (1948; Zbl 0038.03003)] and \textit{P. J. Cohen} [Proc. Natl. Acad. Sci. USA 50, 1143--1148 (1964; Zbl 0192.04401)]. Both of these results assume that the ZFC axioms themselves do not contain a contradiction. The author also deals with Martin's axiom \textit{D. A. Martin} and \textit{R. M. Solovay} [Ann. Math. Logic 2, 143--178 (1970; Zbl 0222.02075)] which states that all cardinals less than \(c\), the cardinality of the continuum, behave like \(\aleph_{0}\); cf. \textit{D. H. Fremlin} [Manuscr. Math. 33, 387--405 (1981; Zbl 0459.28010)].NEWLINENEWLINELet \(G\) denote a locally compact group. \(U\) a unitary representation on a Hilbert space \(H\), \(\mathcal{L}(H)\) the space of bounded linear operators on \(H\) with the weak operator topology. Measurability is understood here in the sense of belonging to the \(\sigma\)-algebra generated by the family of Borel sets. NEWLINENEWLINEIn the first part of the article the author proves that a Haar measurable map from \(G\) to \(\mathcal{L}(H)\) is weakly (equivalently strongly) continuous. This is clear for separable \(H\) as it concerns mappings to complex-valued functions with no complications because of size. The proof for locally compact groups is based on the result for Polish spaces (cf. \textit{S. Banach} [Théorie des opérations linéaires. Warszawa, Mathematisches Seminar der Univ. Warschau (1932; JFM 58.0420.01)], who proved that measurable maps between Polish spaces are continuous). \textit{J. Brzuchowski} et al. [Bull. Acad. Pol. Sci., Sér. Sci. Math. 27, 447--448 (1979; Zbl 0433.28001)], proved that if \(\mathcal{A}\) is a point-finite family of null sets with non-null union in a Polish space then there is a subfamily of non-measurable functions of \(\mathcal{A}\) with non-measurable union. The author's Lemma 1.4 is a generalisation of their work to a \(\sigma\)-compact locally compact group \(G\) of cardinality not more than \(c\). NEWLINENEWLINEOne calls \(U\) weakly operator measurable if \(U^{-1}(V)\) is measurable for every open set \(V\) in \(\mathcal{L}(H)\). The author proves in Theorem 1.5 that every weakly operator measurable unitary representation of \(G\) is continuous. The proof involves use of the fact that every locally compact group has an open pro-Lie subgroup. NEWLINENEWLINEThe author remarks that this result is known within ZFC but it is not possible to replace the assumption that the family \(\mathcal{A}\) is point-finite even by that \(\mathcal{A}\) is point-countable and comments also on the difference between the approaches to Haar measure due to E. Hewitt and K. A. Ross (from outer regular measures) and of D. H. Fremlin (from inner regular measures) but says that her Theorem 1.5 is valid under both approaches. The reviewer will for brevity not refer the Hewitt-Ross locally null sets. NEWLINENEWLINEIn the second part of the article the author considers automatic continuity, viz. that every measurable homomorphism from a locally compact group to an arbitrary topological group is continuous (cf. \textit{B. J. Pettis} [Ann. Math. (2) 52, 293--308 (1950; Zbl 0037.30501)]) that any Baire measurable homomorphism from a Polish group to a separable group is continuous. The strategy is to use some sort of measurability condition to ensure automatic continuity. NEWLINENEWLINEThe author calls \(A \subset G\) extra-measurable if \(SA\) is measurable for every \(S\subset G\). However the author's Theorem 2.4 shows that extra-measurable sets do not exist under Martin's axiom. Theorem 2.1 claims that the existence of discontinuous measurable homomorphisms from \(G\) to a topological group \(H\) implies existence of null non-empty extra-measurable sets. However \(H\) is not arbitrary but constructed from the topological group \(G\) by changing the basis of neighborhoods of the identity in \(G\) to be the family specified in (2.1). If there is a continuous homomorphism \(G \to H\) which is measurable and discontinuous then conditions (2.1) guarantee that the basis can be made up of non-empty null extra-measurable sets. In the case of Abelian \(G\), automatic continuity should be equivalent to the existence of this sequence of null extra-measurable sets (Proposition 2.2). NEWLINENEWLINEThe author states that by Theorem 2.4 it is consistent with ZFC that a nonempty null set cannot be extra-measurable, so it is likewise consistent that every measurable homomorphism from a locally compact group to any topological group is continuous (but here Martin's axiom has been added). NEWLINENEWLINEFor automatic continuity of group homomorphisms for Polish groups Haar measurability may be replaced by universal measurability. A subset \(A\) of a Polish space \(X\) is universally measurable if it is measurable with respect to every complete probability measure on \(X\) that measures all Borel subsets of \(X\).NEWLINENEWLINE\textit{A. Kleppner} [Proc. Am. Math. Soc. 106, No. 2, 391--395 (1989; Zbl 0693.22004)] obtained an automatic continuity for homomorphisms based on the Hewitt-Ross Haar measure. He proved existence of extra-measurable sets, postulating a `semimeasurability' of the mapping \(G \to H\) in order to have Borel sets in \(G\) which correspond to Borel sets in \(H\).NEWLINENEWLINEAssuming Martin's axiom the author obtains the following results (referring to a communication with A. Kharazishvili): for a non-empty null set \(A\) in a locally compact Polish group there is a set \(S \subset G\) such that \(SA\) is nonmeasurable. Theorem 2.4 extends this to a locally compact \(G\) (making use of an open pro-Lie subgroup in \(G\)). The author claims automatic continuity of homomorphisms \(G \to H\) follows by combining Theorems 2.1 and 2.4 but the reviewer is unhappy about the author's use of Thorem 2.1 for an arbitrary topological group.NEWLINENEWLINEAmong the final remarks the author defines \(S \subset G\) to be small if the union of every family of translates of \(S\) of cardinality less than \(c\) is null. Martin's axiom guarantees that every null set is small.