It is well known that if $\langle f_n:n\in\mathbb{N}\rangle$ is a sequence of continuous functions, $f_n\colon\mathbb{R}\to\mathbb{R}$, then $\big\{x\in\mathbb{R}:\lim_{n\to\infty}f_n(x)\text{ exists}\big\}$ is an $F_{\sigma\delta}$-set (see this post). I am asking if the converse is true, i.e., whether for every $F_{\sigma\delta}$-set $E\subseteq\mathbb{R}$ there exists a sequence $\langle f_n:n\in\mathbb{N}\rangle$ of continuous functions, $f_n\colon\mathbb{R}\to\mathbb{R}$, such that $\big\{x\in\mathbb{R}:\lim_{n\to\infty}f_n(x)\text{ exists}\big\}=E$.
My attempt: I would try to prove it in two steps.
(1) Given an $F_{\sigma\delta}$-set $E$, find closed sets $E^k_n$, $n,k\in\mathbb{N}$, such that $E^k_n\supseteq E^l_n$ and $E^k_n\subseteq E^k_m$ for $k\le l$ and $n\le m$, and $E=\bigcap_k\bigcup_n E^k_n$.
(2) Given $E^k_n$ as above, find continuous functions $f_n\colon\mathbb{R}\to\mathbb{R}$ such that for every $x$, $x\in E^k_N$ iff $\left|f_n(x)-f_m(x)\right|\le 2^{-k}$ for all $m\ge n\ge N$.
(1) would be accomplished as follows. Let $E=\bigcap_k\bigcup_n F^k_n$, $F^k_n$ closed. Let $\langle G^0_n:n\in\mathbb{N}\rangle$ consists of all elements of $\langle F^0_n:n\in\mathbb{N}\rangle$, each repeating infinitely many times. Let $\langle G^1_n:n\in\mathbb{N}\rangle$ consists of all possible intersections $F^0_i\cap F^1_j$, $i,j\in\mathbb{N}$, each repeating infinitely many times and ordered so that $G^1_n\subseteq G^0_n$ for every $n$. Similarly, let $\langle G^k_n:n\in\mathbb{N}\rangle$ consists of all possible intersections $G^0_{i_0}\cap\cdots\cap G^k_{i_k}$, $i_0\dots,i_k\in\mathbb{N}$, each repeating infinitely many times and ordered so that $G^k_n\subseteq G^l_n$ for every $n$, whenever $l<k$. This way we obtain $\bigcup_n G^k_n=\bigcap_{l\le k}\bigcup_n F^l_n$. Finally, put $E^k_n=\bigcup_{m\le n}G^k_m$. Then $\bigcup_n E^k_n=\bigcup_n G^k_n$ and hence $\bigcap_k\bigcup_n E^k_n=\bigcap_k\bigcup_n G^k_n=\bigcap_k\bigcup_n F^k_n=E$.
Is that correct? And can (2) be accomplished, too?
Best Answer
As it was mentioned here, the statement in question was proved by Hahn in 1919 and independently by Sierpinski in 1921. The proof can be found also in Hausdorff's book Set Theory and in Kechris' Classical Descriptive Set Theory. The most accessible proof is that from Kechris' book, and is essentially the same as Hausdorff's one. Original Hahn's proof uses a similar argument, but is slightly more difficult to follow. Sierpinski's proof is based on a completely different idea and I had a problem to understand it. You can find its outline in the above referenced discussion. I will sketch the Kechris' version of the proof here, for those who do not want to look for it in the book.
An important tool used in the proof is the following lemma due to Baire.
The most elegant way to prove the lemma is to use Yosida regularization, namely, to define $f_n(x)=\inf\{f(y)+n\lvert x-y\rvert:y\in\mathbb{R}\}$.
Then we can prove a special case of the theorem.
Proof. Assume that $E=\bigcup_n E_n$, where $E_n$ is closed and $E_n\subseteq E_{n+1}$, for every $n$. Then the function $f:\mathbb{R}\to[0,\infty]$ defined by $f(x)=\min\{n:x\in E_n\}$ if $x\in E$, and $f(x)=\infty$ for $x\notin E$, is lower semicontinuous. By the lemma, there exists a nondecreasing sequence $\langle f_n:n\in\mathbb{N}\rangle$ of continuous functions converging to $f$. We may also assume that for every $x$, $\lvert f_{n+1}(x)-f_n(x)\rvert\le 1/2$. This can be achieved by taking $f'_n(x)=\min\{n,f_n(x)\}$ and then inserting some terms into the sequence $\langle f'_n:n\in\mathbb{N}\rangle$. Finally, we put $g_n(x)=\sin(\pi f_n(x))$. If $x\in E$ then $f_n(x)$ converges to some integer and hence $g_n(x)$ converges to $0$. If $x\notin E$ then $f_n(x)$ converges to $\infty$, and since $\lvert f_{n+1}(x)-f_n(x)\rvert\le 1/2$, it is clear that $g_n(x)$ oscillates. This finishes the proof of the special case.
To finish the proof of the theorem, assume that $E=\bigcap_k E_k$, $E_k$ being $F_\sigma$. For every $k$, there exists a sequence $\langle g^k_n:n\in\mathbb{N}\rangle$ of continuous functions, $g^k_n\colon\mathbb{R}\to [-1,1]$, such that $g^k_n(x)\to 0$ if $x\in E_k$, and $g^k_n(x)$ diverges if $x\notin E_k$. Let us take a sequence consisting of all functions $2^{-k}g^k_n$, for $k,n\in\mathbb{N}$. As it can be easily checked, the obtained sequence has the desired properties.
It seems that it would be quite difficult to rewrite this proof into a direct definition of the sequence $\langle f_n:n\in\mathbb{N}\rangle$ from the sets $E^k_n$ mentioned in the question. Also, in my unsuccessful attempts I was trying to use only the normality of $\mathbb{R}$ and the fact that open sets of reals are $F_\sigma$. However, the proof of the Baire's statement depend heavily on the metric, so I was trying probably to prove also a kind of metrization theorem. It would be interesting to know if the result holds also in non-metrizable spaces.