Normed versus topological groups: Dichotomy and duality

114 N. H. Bingham and A. J. OstaszewskiTheorem 11.5 (Nikodym’s Theorem – [Nik]; [Jay-Rog, p. 42]). The Baire sets of a spaceX are closed under the Souslin operation. Hence Souslin-F(X) sets are Baire.We promised examples of Baire sets; we can describe a hierarchy of them.Examples of Baire sets. By analogy with the projective hierarchy of sets (known alsoas the Luzin hierarchy – see [Kech], p. 313, which may be generated from the closed setsby iterating projection and complementation any finite number of times), we may formthe closely associated hierarchy of sets starting with the closed sets and iterating anyfinite number of times the Souslin operation S (following the notation of [Jay-Rog]) andcomplementation, denoted analogously by C say. Thus in a complete metric space oneobtains the family A of analytic sets, by complementation the family CA of co-analyticsets, then SCA which contains the previous two classes, and so on. By Nikodym’s theoremall these sets have the Baire property. One might call this the Souslin hierarchy.One may go further and form the smallest σ-algebra (with complementation allowed)closed under S and containing the closed sets; this contains the Souslin hierarchy (implicitthrough an iteration over the countable ordinals). Members of the latter family arereferred to as the C-sets – see Nowik and Reardon [NR].Definitions. 1. Say that a function f : X → Y between two topological spaces is H-Baire, for H a class of sets in Y, if f −1 (H) has the Baire property for each set H in H.Thus f is F(Y )-Baire if f −1 (F ) is Baire for all closed F in Y. Taking complements, sincef −1 (Y \H) = X\f −1 (H),f is F(Y )-Baire iff it is G(Y )-Baire, when we will simply say that f is Baire (‘f has theBaire property’ is the alternative usage).2. One must distinguish between functions that are F(Y )-Baire and those that lie in thesmallest family of functions closed under pointwise limits of sequences and containing thecontinuous functions (for a modern treatment see [Jay-Rog, Sect. 6]). We follow traditionin calling these last Baire-measurable (originally called by Lebesgue the analyticallyrepresentable functions, a term used in the context of metric spaces in [Kur-1, 2.31.IX,p. 392] ; cf. [Fos]).3. We will say that a function is Baire-continuous if it is continuous when restricted tosome co-meagre set. In the real line case and with the density topology, this is Denjoy’sapproximate continuity ([LMZ, p. 1]); recall ([Kech, 17.47]) that a set is (Lebesgue)measurable iff it has the Baire property under the density topology.The connections between these concepts are given in the theorems below. See thecited papers for proofs, and for the starting point, Baire’s Theorem on the points ofdiscontinuity of a Borel measurable function.