About Countable union of Baire class 2 functions

Question

I Just wanted to say I do not know much about the Baire class 2 functions this why I am asking this question. For each $n\in\Bbb Z$, let $f_n\colon [n,n+1]\to\Bbb R$ be a function with Baire class 2 such that $f_{n}(n+1)=f_{n+1}(n+1)$. Now, define $f$ as follows $$f=\bigcup_{n\in\Bbb Z} f_n.$$ Clearly, $f$ is a function form $\Bbb R$ to $\Bbb R.$ Is $f$ still in Baire class $2$? Any help will be appreciated greatly.

Answer 1

In fact, a more general result holds:

Suppose $f_i:[i,i+1]\rightarrow\mathbb{R}$ are Baire class $\alpha$ for $i\in\mathbb{N}$, with $f_i(i+1)=f_{i+1}(i+1)$. Then $\bigcup_{i\in\mathbb{N}}f_i:[0,\infty)\rightarrow\mathbb{R}$ is also Baire class $\alpha$.

The proof is by induction. The base case $\alpha=0$ corresponds to gluing together continuous functions, so is trivial. Here's a sketch of how to handle the induction step (there are a couple issues with it which I mention below):

Let $(g^i_j)_{j\in\mathbb{N}}:[i,i+1]\rightarrow\mathbb{R}$ be a sequence of functions approaching $f_i$ pointwise, each of which is Baire class $<\beta$. We want to "glue together" these approximations to get a sequence of approximations for the union of the $f_i$s. One natural idea here is to let $h_j=\bigcup_{i\in\mathbb{N}}g_j^i$. There are a couple problems with this however:

• Although the $f_i$s "agree at the endpoints," the $g_j^i$s might not: we need not have $g_j^i(i+1)=g_j^{i+1}(i+1)$. That is, $h_j$ as defined above may not be a function.

• Even if each $g^i_j$ is simple their union need not be in case $\beta$ is a limit ordinal. For example, what if $\beta=\omega$ and $g^i_j$ is Baire class $i$? Then the $h_j$ defined above is Baire class $\omega$, which is too big.

Fixing these issues is a good exercise. The significant one is really the second bulletpoint, for which I'll give a hint:

Redefine $h_j$ to only take into account the $g_j^i$s with $i\le j$. Basically, for $x>j+1$ we'll want $h_j(x)$ to be "boring." This won't affect the pointwise limit of the $h_j$s and will let us control their complexity.