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If $m$ is an outer measure on a set $X$, a subset $E$ of $X$ is called $m$-measurable iff $$ m(A) = m(A \cap E) + m(A \cap E^c) $$ for all subsets $A$ of $X$.

The collection $M$ of all $m$-measurable subsets of $X$ forms a $\sigma$-algebra and $m$ is a complete measure when restricted to $M$.

Is $M$ the largest $\sigma$-algebra on $X$ on which $m$ is a measure (i.e., on which $m$ is countably additive)? If not, what is?

Is $M$ the largest $\sigma$-algebra on $X$ on which $m$ is a complete measure? If not, what is?

I am especially interested in the case when $X$ is $\mathbb{R}$ or $\mathbb{R}^n$ and $m$ is the Lebesgue outer measure. In this case $M$ is the Lebesgue $\sigma$-algebra.

ADDED:

Julián Aguirre (thanks!) has shown in his response below that the answer to the first question is yes when $X$ is $\mathbb{R}^n$ and $m$ is the Lebesgue outer measure. Hence the answer to the second question in this situation is also yes.

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