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We define a set $E \subseteq X $ as measurable if $$l(A)=l(E \cap A)+l(E^{c} \cap A)$$ for every $A \subseteq X $, where $l$ is an outer measure. Suppose now we have two measures $\mu_{1}$ and $\mu_{2}$ such that for every $E \subseteq X$ $$\mu_{1}(E)=0 \iff \mu_{2}(E)=0$$ Does this imply that the two sigma algebras of measurable sets related to the measures above by Carathéodory's Extension Theorem are equal?

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