On all textbooks that I have read so far, there are only two conditions for which a function $\mu:\Sigma\to[0,+\infty]$ is a measure on a measurable space $(X,\Sigma)$, which are:
However, most of the textbooks leave the non-negativity of $\mu$ as a remark. Why is that not included in the definition? If that follows immediately from the definition, then how?
On
The question is not clear, because you don't specify exactly what definition you have in mind (the two major typos don't help).
A version of the standard definition is this:
Def A measure on the measurable space $(X,\Sigma)$ is a map $\mu:\Sigma:\to[0,\infty]$ such that $\mu(\emptyset)=0$ and $\mu$ is $\sigma$-additive.
Non-negativity is included in that definition! Saying $\mu:\Sigma\to[0,\infty]$ means precisely that $0\le \mu(E)\le\infty$ for every $e\in\Sigma$.
There are various types of measures: positive measures, real measures, complex measures, vector measure etc. In elementary books dealing with positive measures they impose the condition $\mu(E) \geq 0$ for all sets $E$ in the sigma algebra. Note that if $m$ is Lebesgue measure them $\mu(E)=-m(E)$ defines a set functions satisfying 1) and 2). So positivity does not follow from 1) and 2).