Decomposing Borel Sets and Functions and the Structure of Baire Class 1 Functions

All spaces considered are metric separable and are denoted usually by the letters X, Y , or Z. ω stands for the set of all natural numbers. If a metric separable space is additionally complete, we call it Polish; if it is a continuous image of ω or, equivalently, of a Polish space, it is called Souslin. The main subject of the present paper is the structure of Baire class 1 functions. Recent developments in Banach space theory, in particular discoveries of Rosenthal, and Bourgain, Fremlin, and Talagrand (see [R] for a survey of these developments), stimulated investigations into the structure of Baire class 1 functions. The interested reader may consult papers by Haydon, Odel, Rosenthal [HOR], Kechris, Louveau [KL], Rosenthal [R2], and references quoted there. (These investigations have already provided new applications in Banach space theory; see, for example, [R1].) In our study of Baire class 1 functions, we will be interested in two decomposition properties, one of them defined by Lusin, the other one by Jayne and Rogers. First, however, we want to consider a more general problem of determining how difficult it is to represent a Borel set as a union of simpler Borel sets or the graph of a Borel function as a union of the graphs of simpler Borel functions. Using Effective Descriptive Set Theory, in particular Louveau’s theorem, we show that if A ⊂ X is Borel, X Polish, then A ∈ Σα or there is a continuous injection φ : ω → A such that φ−1(B) is meager for any B ⊂ A which is Σα. This gives a new proof of J. Stern’s result that if a Borel set A is the union of < cov(M) sets in Σα, then A is itself Σα. (cov(M) is the smallest cardinality of a family of meager sets covering R.) We prove similar results for functions. Put, for f : X → Y and a family of functions G, dec(f,G) = min{|F| : ⋃ F = X, ∀Z ∈ F f |Z ∈ G}. We study these coefficients for various G, in particular for G = Baire class α functions. We show, for example, that given f : X → Y Borel, X Polish, and α < ω1, either dec(f, Baire class α) ≤ ω or there is a continuous injection φ : ω → X such that φ−1(A) is meager for any A ⊂ X with f |A in Baire class α; thus, in the latter case, dec(f, Baire class α) ≥ cov(M). These results imply that the decomposition coefficients defined in [CMPS] and proved there to be > ω are actually ≥ cov(M).