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?