Please forgive my (lack of) LaTeX. I am a bit confused by the matter of completeness in the first half of Prop 1.14 in Folland's Real Analysis. The setup is that we have a premeasure $\mu_0$ defined on an algebra $A$, and the claim is that there exists a measure $\mu$ on the $\sigma$-algebra $M(A)$ generated by that algebra, and that $\mu$ is equivalent to $\mu_0$ on $A$. It also mentions that this is in fact ∗ (the outer measure induced by $_0$) restricted to $M(A)$.
Prop 1.13 shows that every set in the algebra $A$ on which a premeasure $\mu_0$ is defined is $\mu∗$-measurable by the outer measure $\mu∗$ induced by that $\mu_0$. It also shows that $\mu∗$ and $\mu_0$ agree on $A$.
Carathéodory's Theorem shows that an outer measure $\mu∗$ is complete on $M∗$, the $\sigma$-algebra of $\mu∗$-measurable sets. So please correct me where I am wrong:
Since every set in $A$ is $\mu∗$-measurable (1.13), then $M∗$ contains $M(A)$. Since $\mu∗$ is complete on $M∗$ (Carathéodory) then $\mu = \mu∗$ must be complete on $M(A)$ as well. Finally, since $\mu∗$ agrees with $\mu_0$ on $A$ (1.13), we have a measure $\mu$ on $M(A)$ that is equal to $\mu∗$ on $M(A)$, and whose restriction to $A$ is $\mu_0$, as claimed by 1.14 -- and this measure is complete on $M(A)$.
Sorry for the confusing wording. Folland does not state that the measure is complete, so have I made a mistake in my assumptions or reasoning? Thanks in advance for any replies