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A Baire 1 function on the reals is the pointwise limit of a sequence of continuous functions. Assuming a bounded Baire 1 function on the unit interval, can we say anything about the modulus of convergence? In particular does such a modulus always belong to a certain (nice) space?

(PS: to be absolutely clear, let $f:[0,1]\rightarrow [0,1]$ be a given Baire 1 function, i.e. there is a sequence $(f_n)_{n \in \mathbb{N}}$ of continuous function $f_n:[0,1]\rightarrow [0,1]$ such that for all $x\in [0,1]$, we have $$ (\forall x \in [0,1], \epsilon>0)(\exists n)(\forall m\geq n)(|f_m(x)-f(x)|<\epsilon). $$ A modulus of convergence is any function $\Psi:[0,1]^2 \rightarrow \mathbb{N}$ sich that $\Psi(x,\epsilon)$ is a number $n$ as in the above formula.

Sam Sanders
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  • Possibly relevant is Eliakim Hastings Moore's notion of relatively uniform convergence, introduced by Moore around 1908 (see footnote 86 on p. 64 of this paper). This notion was used/studied by a few people during the next 20 years or so, after which it pretty much faded into mathematical oblivion (see my comment to this mathoverflow question), (continued) – Dave L Renfro Sep 14 '22 at 05:53
  • although the google search I gave just above shows that some papers in the last few years have made use of it. An idea I had in the mid 1990s, but thus far haven't investigated so I'll throw it out here for anyone who might be interested, was to investigate this notion in the following way. For a sequence of continuous functions on $[0,1],$ if pointwise convergent then the limit function is continuous on a co-meager set and if uniformly convergent then the limit function is continuous on a (the) co-empty set. Since every meager set is a $\sigma$-$h$-porous (continued) – Dave L Renfro Sep 14 '22 at 05:54
  • for some scale function $h$ (same as a Hausdorff measure scale/gauge function -- see Notes 5, 6, 7 here), perhaps one can prove a result along the lines of "relatively uniformly convergent to some specified scale (in Moore's sense) implies there exists a scale function $h$ (depending in some way on Moore's scale) such that the limit function is continuous on a co-$\sigma$-$h$-porous set" (for some notion of lower or upper porosity, such as mentioned at the end of this MSE answer). – Dave L Renfro Sep 14 '22 at 05:54
  • Potentially relevant is the notion of convergence rank of a Baire class 1 function, see for example Kechris-Louveau A classification of Baire class 1 functions – Alessandro Codenotti Sep 14 '22 at 13:26
  • If you want to go down the rabbit hole suggested by @Alessandro Codenotti, then the list of items I gave in this mathoverflow question will be useful. I've probably dug enough for you to go down the rabbit hole I suggested earlier :) – Dave L Renfro Sep 14 '22 at 14:00
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    @DaveLRenfro thanks for the list! This is going to be useful for me – Alessandro Codenotti Sep 14 '22 at 16:18

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