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Let $(X,M,\rho)$ be a finite measure space. Suppose $U \subset M$ is an algebra of sets and $\mu: U \longrightarrow \mathrm{C}$ is a complex, finitely additive measure such that $|\mu(E)| \leq \rho(E) < \infty$ for all $E \in U.$ Show that there is a complex measure $\nu: M \longrightarrow \mathrm{C},$ whose restriction to $M$ is $\mu$, and such that $|\nu(E)| \leq \rho(R)$, for all $E \in M$.

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Which results about extension of a measure do you know? –  Davide Giraudo Oct 16 '12 at 19:50
No one @DavideGiraudo. –  Kelson Vieira Oct 20 '12 at 1:09
@DavideGiraudo This doesn't fit into the framework of the Caratheodory theorem (which involves extending a positive premeasure as I know it). –  user44532 Dec 30 '12 at 3:33

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