real analysis - Second Baire class and Borel measurable

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?