Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$

Question

A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' Classical Descriptive Set Theory, p. 192):

Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\boldsymbol{\Sigma}_2^0$-measurable function, then $f$ is the pointwise limit of a sequence of continuous functions.

The theorem can be extended to functions with codomain $\mathbb{R}^n$ for each $n\in\mathbb{N}$. My questions are:

1. Does this theorem hold for functions with codomain $\mathbb{R}^\omega$?
2. Is there a counterexample of a $\boldsymbol{\Sigma}_2^0$-measurable function $f:\mathbb{R}^\omega\rightarrow\mathbb{R}^\omega$ which is not pointwise limit of a sequence of continuous functions?

Thanks!

Answer 1

Is it indeed the case that the statement can be extended to functions with codomain $\mathbb{R}^\omega$.

Suppose we have $f:\mathbb{R}^\omega\rightarrow\mathbb{R}^\omega$ which is $\boldsymbol{\Sigma}_2^0$-measurable, then in particular the functions $f^n:\mathbb{R}^\omega\rightarrow\mathbb{R}$ with $f^n(x)=f(x)(n)$ are $\boldsymbol{\Sigma}_2^0$-measurable.
Applying Lebesgue, Hausdorff and Banach theorem we have that for each $n$ there exists a sequence $(g_m^n)_{m\in\omega}$ of functions from $\mathbb{R^\omega}$ into $\mathbb{R}$ such that $g_m^n \longrightarrow g^n$ pointwise as $m\rightarrow \infty$.
Now if we consider the sequence $(g_m)_{m\in\omega}$ of functions functions from $\mathbb{R^\omega}$ into $\mathbb{R}^\omega$ with $g_m(x) = (g_m^n(x))_{n\in\omega}$, then we have that they are all continuous and they converge pointwise to $f$.