How much we can extend meaurable sets?

Question

Does there exist a $\sigma$-algebra $A \supset M$ (with a measure $m$ defined on it) such that

1. $m(I) = L(I)$ where $I$ is any interval in $\Bbb{R}$ and $L(I)$ means the length of the interval,
2. $m$ is $\sigma$-additive.
3. $m$ is translation invariant,

where $M$ is the set of all measurable sets with respect to Lebesgue outer measure (i.e. the usual class of Lebesgue-measurable sets).

Answer 1

This was first done by Szpilrajn (1935) and later strengthened to non separable extensions by Kakutani and Kodaira (1950). Ciesielski gives a short proof of the non existence of any maximal such extension here.