A criterion for measurability of countable-to-one functions (Q1824712)

A subset of R is analytic if it is the image of a Borel subset of R under a measurable map. The author proves: Let f: \(X\to Y\) be a one-one correspondence between subsets X and Y of R. Suppose that X is analytic. In order that f be a Borel-isomorphism, it is necessary and sufficient that for each \(A\subset X\), the sets A and f(A) be Borel-isomorphic. The author proves another theorem from which it follows that under the continuum hypothesis, the condition of analyticity in the above theorem is not needed.