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!



Baire class $1$ functions and Baire's characterization theorem
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](https://www.tau.ac.il/%7Etsirel/Courses/MeasCategory/lect5.pdf) 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).


