Added that "g" has to be measurable, which we never said, but we wanted. If g need not be measurable then the conditions on the space X become much weaker, on the other hand. Also, swapped order of the last two paragraphs and added necessary condition of completeness to the Kuratowski comment.

Fixed typo "supose" => "suppose" and added parenthetical (remark) about the sufficiency of separability and completeness, since this raises the question as to whether there's a Borel mapping if X is merely metrizable, which is the setting of Kuratowski.

(sorry. adding logic to prevent someone making the same mistake i did, which now appears as a comment below. if someone could add a comment that I now agree this is a counterexample, I'd appreciate that!)

added polish suggestion by Jason Rute

added 489 characters in body

Put a brief version of the question at the start so that people browsing in Questions can see it without clicking through.