1
$\begingroup$

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 ${\cal A}'\subseteq{\cal A}$?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.