possibility of extension of Lesbegue measure on the real line

Question

The Lesbegue measure on the real line is defined on the class of Lesbegue-measurable sets, whose elements satisfy the Caratheodory condition.

Here what I am curious about is that if it is possible to 'extend' the domain of Lesbegue measure to some properly bigger subset of the real line's power set and define a measure on the set which is just the Lesbegue measure when restricted to the domain of Lesbegue measure and satisfies the countable additivty.

If, not possible, could anyone explain the reason?

Answer 1

You'd have to lose something that makes the Lebesgue measure what it is.

The Lebesgue measure has a few distinguishing properties: 1) Countable additivity 2) Preservation of measure under congruences (translations, rotations, reflections) 3) Volume of the unit cube is 1 4) Complete-ness

The Lebesgue measure is unique by uniqueness of extension of Borel measure to a complete measure.