# Examples of functions that do not belong to any Baire class

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.

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 $g (r)=f (r)(r)+1.$

The Baire functions (starting with $$\mathcal{C}_{00}(\mathbb{R})$$ or $$\mathcal{C}_b(\mathbb{R})$$), or rather, the union of all Baire classes is the family of Borel measurable functions. Any Lebesgue-measurable function that is not Borel will do (The axiom of choice plays a role here). See for example Cohn. D., Measure Theory, 2nd edition, Birkhäuser, 2013, pp. 48 for a concrete example.

Here is a brief explanation of how the collection of all Baire functions is done thorough a transfinite inductive procedure:

Suppose $$\mathcal{V}$$ is a collection of real valued functions defined an a set $$\Omega$$. $$\mathcal{V}$$ is said to be sequentially closed if for any convergent sequence $$(f_n:n\in\mathbb{N})\subset \mathcal{V}$$, $$f:=\lim_nf_n\in\mathcal{V}$$. It is easy to check that the intersection of sequentially closed families is sequentially closed: suppose $$\mathfrak{F}$$ is a collection of sequentially closed families. If $$(f_n:\in\mathbb{N})\subset\bigcap\mathfrak{F}$$ and $$f=\lim_nf_n$$ exists, then for any $$\mathcal{F}\in\mathfrak{F}$$, $$(f_n:n\in\mathbb{N})\subset\mathcal{F}$$ and so, $$f\in\mathcal{F}$$.

Suppose $$\mathcal{E}$$ is a collection of real-valued functions defined on a set $$\Omega$$, let $$\mathcal{E}^\sigma$$ denote the smallest sequentially closed family that contains $$\mathcal{E}$$. This family exists from the observation above and since $$\mathbb{R}^\Omega$$ is clearly sequentially closed and contains $$\mathcal{E}$$.

Now, consider the following transfinite induction procedure: Let $$\mathcal{E}_0=\mathcal{E}$$; if $$\mathcal{E}_\alpha$$ has been defined for all ordinals $$\alpha<\beta$$, define $$\mathcal{E}_\beta$$ as \begin{align} \mathcal{E}_\beta=\left\{\begin{array}{lcr} \{f\in\mathbb{R}^\Omega: f=\lim_nf_n\,\text{for some}\,(f_n:n\in\mathbb{N})\subset\mathcal{E}_\alpha\} &\text{if}& \beta=\alpha+1\\ \bigcup_{\alpha<\beta}\mathcal{E}_\alpha &\text{if}& \beta\neq\alpha+1 \end{array} \right. \end{align} That is, if $$\beta$$ is the successor of $$\alpha$$, then $$\mathcal{E}_\beta$$ is the collection of pointwise limits of sequences in $$\mathcal{E}_\alpha$$, and if $$\beta$$ is not the successor of $$\alpha$$, then $$\mathcal{E}_\beta=\bigcup_{\alpha<\beta}\mathcal{E}_\alpha$$. It follows by construction that $$\mathcal{E}_\beta\subset\mathcal{E}^\sigma$$ for any ordinal $$\beta$$. The collection $$\mathcal{E}_\beta$$ is the $$\mathcal{E}$$-Baire familly of class $$\beta$$. (The case $$\mathcal{E}=\mathcal{C}_b(\mathbb{R})$$ yields the commonly known Baire classes).

It is easy to check that this construction stabilizes after the first uncountable ordinal $$\omega_1$$, for $$\mathcal{E}_{\omega_1}$$ is sequentially closed: suppose $$(f_n:n\in\mathbb{N})\subset \mathcal{E}_{\omega_1}$$, and $$f_n\xrightarrow{n\rightarrow\infty} f$$ pointwise to some function $$f\in\mathbb{R}^{\Omega}$$. Then there is a sequence of countable ordinals $$\alpha_n$$ such that $$f_n\alpha_n$$. As $$\bigcup_n[0,\alpha_n]$$ is countable and contained in $$P_{\omega_1}:=[0,\omega_1)$$, there is $$\gamma\in P_{\omega_1}$$ such that $$\alpha_n\leq \gamma$$ for all $$n\in\mathbb{N}$$; hence, $$(f_n:n\in\mathbb{N})\subset \mathcal{E}_{\gamma}$$ and so, $$f\in \mathcal{E}_{\gamma+1}\subset\mathcal{E}_{\omega_1}$$.

• This statement appears (without proof) in Bichteler, K., Stochastic Integration with Jumps, Combridge U. Press, 2002., pp. 392., and no special structure on $$\mathcal{E}$$ (Stone lattice and/or ring for example) is assumed.

• The special case $$\mathcal{E}=C_b(\mathbb{R})$$ for instance yields that the space $$\mathscr{M}(\mathbb{R})$$ of Borel measurable functions is the same as $$(\mathcal{C}_b(\mathbb{R}))_{\omega_1}$$. Indeed, clearly $$\mathcal{E}^\sigma$$ is contained in the space of all real-valued measurable functions $$\mathscr{M}(\mathbb{R})$$. Notice that the indicator function of any bounded open interval $$I$$ is in $$\mathcal{E}_1$$. The collection of sets in $$\mathbb{R}$$ whose indicator functions are in $$\mathcal{E}_{\omega_1}$$ is a $$\sigma$$-algebra; Hence $$\mathscr{M}(\mathbb{R})\subset\mathcal{E}^\sigma$$.