What is restriction of a measure to an algebra? I am reading Folland's Real analysis book and I saw following sentence in the Theorem 1.14:


  Let $A 
\subset \mathcal{P}(X)$ be an algebra, $\mu_0$ a premeasure on $\mathcal{A}$, and $\mathcal{M}$ the $\sigma$-algebra generated by $\mathcal{A}$. There exists a measure $\mu$ on $\mathcal{M}$ whose restriction to $\mathcal{A}$ is $\mu_0$ -- namely, $\mu = \mu^* | \mathcal{M}$ (where $\mu^*$ is outer measure).


What I want to ask is, what book means by saying  There exists a measure $\mu$ on $\mathcal{M}$ whose restriction to $\mathcal{A}$ is $\mu_0$?
There exists a measure $\mu$ on $\mathcal M$ that gives you the same value for each element of $\mathcal A$ as $\mu_0$ would give you. That's all.
This means that if you define $ \mu(E) = \mu^*( E) $ for all $ E  \in  \mathcal{M} $, then $ \mu $ is a measure on $\mathcal{M}$.