Let $\mu$ be a $\sigma$-finite pre-measure a semi-ring and let ${\cal A}$ denote the $\sigma$-algebra of Caratheodory-measurable sets. I understand that for $D\notin {\cal A}$ one can construct an extension of $\mu$ on a $\sigma$-algebra containing ${\cal A}\cup\{D\}$ but the extension will not be unique in general. Now I ask myself, can one prove that ${\cal A}$ is the largest $\sigma$-algebra on which $\mu$ is uniquely extendable in the sense that if $\mu$ is uniquely extendable on ${\cal A}'$ then ${\cal A}'\subseteq{\cal A}$?


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