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Let $(X,\Sigma,\mu)$ is a finite signed measure space. Thus $\mu$ can have negative values.

I couldn't prove that if $A \subset B$ then $\mu(A) \leq \mu(B)$ i.e. monotonicity. It is very clear for measures but I think it is not true for signed measures. However I couldn't find a counter example.

If someone can tell me a counter example or hint me about why it is true, I'll be very glad.

Thanks in advance

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2 Answers

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Well, consider a very trivial example like $X=\{0,1\}$, $\Sigma$ the power set and define the measure on singletons by $\mu(\{0\})=1, \mu(\{1\})=-1$. Then take $A = \{0\}$ and $B=X$.

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Take $\mu=\delta_1-2\delta_2$ on $\Bbb{R}$ where $\delta_a$ is the Dirac measure at $a$

$[0,1] \subseteq [0,2]$

$\mu([0,1])=1>\mu([0,2])=-1$

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