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



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> 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.
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> 
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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!



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


