Lets say A and B are two lebesgue measurable subsets of the real line, both of positive measure. Is it always true that there is bi-measurable function f mapping A to B? If so, how do we prove that?
By bi-measurable I mean that there are measurable subsets A’ and B’ such that AA’ and BB’ are null sets and f restricted to A’ is a bijective function from A’ to B’ such that the function and its inverse are both measurable.
What about a general measure space? It’s easy to construct a counterexample using a discrete space with counting measure. However is there anything we can say about when a general measure space does have the given property and when not? What if we additionally assume that A and B have the same measure?
If that makes a difference you may additionally assume at any point that A and B have finite measure.