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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?

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

2 Answers 2

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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).

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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 $e\in\Sigma$.

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