Baire class $1$ functions and Baire's characterization theorem

Question

Kechris in his Classical Descriptive Set Theory book gives the following definition (Definition 24.1) and characterization (Theorem 24.15) of Baire class $1$ functions:

Definition. Let $X,Y$ be metrizable spaces. A function $f:X\rightarrow Y$ is of Baire class $1$ if $f^{-1}(U)$ is $F_\sigma$ for every open set $U\subseteq Y$.

Theorem. (Baire) Let $X$ be a Polish space, $Y$ separable metrizable and $f:X\rightarrow Y$. Then the following are equivalent:

1. $f$ is of Baire class $1$.
2. $f{\upharpoonright} K$ has a point of continuity for every compact $K\subseteq X$.

When I first read this theorem in his book, I suspected its statement differed from Baire's original formulation, but I also thought it was the current and commonly accepted one. But reading some recent papers (mainly in real analysis and general topology) dealing with this class of functions, I discovered that there are a lot of different definitions and characterization theorems in circulation.

For example (I'm leaving out many others) I've seen Baire class $1$ functions being defined as pointwise limits of sequences of continuous functions (this definition and Kechris' are equivalent if $X$ is $0$-dimensional or $Y=\mathbb{R}$), and also I've also read a variety of "Baire's characterization" theorems in which $Y$ is assumed to be some kind of topological vector space. Lastly, I've noticed that every time Baire's characterization theorem is stated differently from Kechris' presentation, it's either without a reference or with a more than hundred years old' one.

Is there a paper giving some semblance of order to all these definitions and theorems going under the same name?

Thanks!

Answer 1

I had a similar problem some time ago and I had the impression that a general review on Baire classes is in fact missing, and would be useful. Probably you already saw it, but in this chapter there is at least a discussion of how the definition through $F_\sigma$ sets originated from the classical definition as pointwise limit, and it covers the characterization theorem.

Other titles I found useful:

• A.C.M. Van Rooij & W.H. Schikhof, A Second Course On Real Functions, Cambridge University Press, 1982.

(If I remember well, the only place where I found an elementary proof that real functions with countably many discontinuities are Baire 1).

• S. Fung, Functions Of Baire Class One, University of California San Diego, Department of Mathematics, May 2013.
• D.M. Bressoud, A Radical Approach to Lebesgue's Theory of Integration

(especially pages 109-119, but in general an interesting reading from a historical point of view).