The following proof is a solution to exercise 1.1.3 of the book “An introduction to measure theory” by Terence Tao.
A box , , is a Cartesian product , where each interval is or or or for with . An elementary set is a finite union of disjoint boxes . Let denote the collection of elementary sets in . The measure is defined as , where denotes set cardinality and .
Let be a function satisfying the non-negativity ( for any elementary set ), finite additivity ( for disjoint elementary sets ) and translation invariance ( for any elementary set and any ) properties.
It will be shown that there exists a positive constant such that , i.e. the functions and are equal up to a positive normalization constant .
Observe that . Due to translation invariance, . Using finite additivity, it follows that . So, for . Since , it follows that .
This result generalizes to intervals for any rational with and. Since , finite additivity and translation invariance lead to .
It will be shown that the result holds also for intervals for any irrational .
The set of rationals is dense, which means that . For some irrational and for each , set , so . Pick some with . For all with , it holds that , whence . So, with , it is true that and consequently .
For any two elementary sets , it can be shown that via the equality , non-negativity and finite additivity. Hence, .
Since are rationals, it is deduced that . Thus, .
gives , hence the sandwich theorem yields .
Combining and gives , so for any irrational .
This effectively completes the proof. There remains to show that is true for Cartesian products of unequal intervals in each coordinate , for any , and for any subinterval of the real line. These are all trivial given the existing foundations.