The existence of a measure-preserving bijection from a unit square to a unit segment

In this paper, we prove the existence of a measure-preserving bijection from unit square to unit segment. This bijection is also called the probability isomorphism between two probability spaces. Then we give a new proof of the existence of the independent random variables on Borel probability space $([0,1],\text{B}([0,1]),\text{Lb})$ that their distribution functions are the given distribution functions.