# Non-negativity of a measure

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:

1. $$\mu(\emptyset)=0$$,
2. $$\mu$$ is $$\sigma$$-additive.

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?

• You included non-negativity when you wrote $\mu : \Sigma \to \color{red}{[0,+\infty]}$ Commented Feb 21, 2019 at 15:47

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).
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 $$e\in\Sigma$$.