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Recently I read this post, which gives examples of Baire class 2 functions. I have also been reading the Wikipedia article on Baire functions. The article claims that "Henri Lebesgue proved that ... there exist functions that are not in any Baire class." I would like to see an example of such a function.

I am having a hard time finding anything that directly addresses my specific question. What I know so far is the definition: such a function is not the pointwise limit of any sequence of Baire class $\alpha$ functions, for any countable ordinal number $\alpha$. I have also found this article, which shows that any such function is not Lebesgue measurable.

I would also like to see some details of Lebesgue's proof that such functions exist. To be more specific, I would like to know whether it is constructive or is only an existence result.

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I believe Lebesgue's proof was nonconstructive, and went as follows. There are at most continuum many countable ordinals, and only continuum many functions in each Baire class (exercise - how many countable subsets does a set of size continuum have?), so only continuum many Baire functions total. But there are two to the continuum many functions. (Note the similarity with the proof that there is a non-Borel set of reals, even a measurable one!)

Meanwhile, I don't think a constructive proof can truly exist, since it is consistent with ZF that every function is Baire. This is the case in models where the reals are a countable union of countable sets.

That said, here is a sorta-example: biject the Baire functions with the reals and diagonalize. That is, let $f $ be a subjection from the reals to the Baire functions, and let $g (r)=f (r)(r)+1. $

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