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

TL;DR

This paper proves the existence of a measure-preserving bijection between a unit square and a unit segment, and provides a new proof for the existence of independent random variables with specified distributions.

Contribution

It establishes a measure-preserving bijection between a square and a segment and offers a novel proof for the existence of independent random variables with given distributions.

Findings

Existence of a measure-preserving bijection from square to segment

New proof for the existence of independent random variables with specified distributions

Reinforces the connection between measure theory and probability distributions

Abstract

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 $([0,1],\B([0,1]),\Lb)$ that their distribution functions are the given distribution functions.