Is Lebesgue Outer Measure Uniquely Characterized by These Requirements?

[Diophantus]

It is a fact that Lebesgue measure is characterised uniquely by the five requirements:

1 - measure of empty set = 0
2 - monotonicity
3 - measure = length for intervals
4 - translation invariance
5 - countable additivity

It is also true that Lebesgue outer measure satisfies:

1 - measure of empty set = 0
2 - monotonicity
3 - measure = length for intervals
4 - translation invariance
5 - countable subadditivity

but I'm dying to know whether these requirements actually characterise Lebesgue outer measure uniquely.

 

[fresh_42]

Hewitt, Stromberg (Real and Abstract Analysis, Springer, GTM 25) don't prove it either, and they are very accurate in those questions, but it is contained as an exercise (12.56). They only require $\mu([0,1])=1$ and translation invariance to conclude $\mu=\lambda$.