Is there only one complete measure extending Borel measure?

Question

Is Lebesgue measure the only complete mesure extending Borel mesure on $\mathbb R$?

If yes, why? If no, what is an example of a complete measure extending Borel measure that is different from Lebesgue measure?

I raise this question following Completion of measure spaces - uniqueness.

Answer 1

And the answer is the same as in the other post. You need to add more conditions to have uniqueness. It is possible to define extensions of the Lebesgue measure, using Carathéodory's theorem, for example.

https://en.wikipedia.org/wiki/Carathéodory%27s_extension_theorem

Then you can take the completion of any of these extended measures, and there you go.