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I am new to Baire class theory, but need it for one part of a project I am working on. I have seen it referenced that functions of second Baire class are Borel measurable. For example here in this book, but I cannot find a proof of this (or have come accross one and not realized I am looking at it).

My other question, regarding this same result, is what what restrictions are there on the domain? For example, if we have a function $f:B \rightarrow \mathbb{R}$, where $B$ is a Borel measurable set in $\mathbb{R}^n$ or any metric space and $f$ is of second Baire class on $B$. Does it follow that $f$ is Borel measurable?

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    $\begingroup$ Any pointwise limit of Borel measurable functions is Borel measurable. It follows that functions of any Baire class are Borel measurable. $\endgroup$
    – Nate Eldredge
    Aug 15, 2015 at 23:57
  • $\begingroup$ Oh, right. Thanks :) That answers my second question too I think. $\endgroup$
    – Ashley
    Aug 16, 2015 at 0:00

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Converting comment to an answer:

Any pointwise limit of Borel measurable functions is Borel measurable. It follows that functions of any Baire class are Borel measurable.

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