Example of a premeasure that is not a measure

I'm reading an example of a premeasure that is not a measure.
We define an algebra on $X = \mathbb{N}$ as the collection of all finite and co-finite sets. We let $\mu$ be a premeasure on this algebra by setting $\mu(E)$ if $|E| = \infty$ and $\mu(E) = |E|$ if $|E| < \infty$. I'm confused about why it's not a measure. We let $(E_{i})_{i=0}^{\infty}$ be a sequence of disjoint sets in $\mathbb{N}$. Then $\mu(\cup_{i=1}^{\infty} E_{i}) = \infty$. And $\sum_{i=1}^{\infty} \mu(E_{i})$ should also be infinity I think because each $\mu(E_{i})$ is positive so the sequence must diverge. So I don't quite see why this $\mu$ is not a measure instead of just a premeasure.