Is Uniqueness Necessary in Mathematical Measures?

[aaaa202]

My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure with the same properties as the Lebesgue measure? What is necessarily so good about the uniqueness property?
Also it has a theorem of its existence. But what does existence of a measure imply? That it is well defined on all sets given the properties that defines it?

 

[Ray Vickson]

My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure with the same properties as the Lebesgue measure? What is necessarily so good about the uniqueness property?
Also it has a theorem of its existence. But what does existence of a measure imply? That it is well defined on all sets given the properties that defines it?

There are lots of other, non-Lebesque measures with the same basic properties, so you need to tell us exactly how the Lebesgue measure differs from these others. We don't know what your book says (and we don't even know the title/author of the book), so you need to fill in the details for us.

 

[aaaa202]

It is "Measures, integrals and Martingales" by Schilling. There exists only one measure, the Lebesgue measure, with the property that the volume of a box in R^n is the product of the length of its sides, which in turn are b-a, b the upper point and a the lower. I just wanted to know why it is so important that it is unique.