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In a number of real analysis texts (I am thinking of Folland in particular), three different kinds of measures are defined.

  • Positive measures: Take values in $[0, +\infty]$
  • Signed measures: Take values in either $(-\infty, \infty]$ or $[-\infty, \infty)$, but cannot assume both $+\infty$ and $-\infty$.
  • Complex measures: Take values in $\mathbb{C}$. Any kind of infinity is not allowed.

My question is: why is this? Is this because of how we set up integrals with respect to these measures, or does it have to do with adding and subtracting measures to make new ones?

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You want to avoid allowing signed measures to take both $\infty$ and $-\infty$ values so you don't have to deal with the ever-thorny question of just what $\infty-\infty$ would mean. Similar difficulties arise with allowing complex valued measures to take the value $\infty$. – Arturo Magidin Jan 30 '11 at 20:58
What Arturo said. $\infty - \infty$ is not just thorny: it's insurmountable. – Pete L. Clark Jan 30 '11 at 21:13
up vote 3 down vote accepted

It has to do with adding measures to satisfy countable summability. Notice that in the complex plane $+\infty$ and $-\infty$ are the same value. So we can't allow one without the other, and who knows what it should mean to add them? On the other hand if we stick to finite values then the summability condition is clear.

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Ah, I see, it has to do with countable additivity. I was wondering where exactly the $\infty - \infty$ problem would arise... – Jesse Madnick Feb 2 '11 at 3:41
@Jesse Madnick: Countable additivity (for the measures of disjoint sets) isn't needed in its full strength to cause ill-definition, just adding measures of two disjoint sets, provided one has $+\infty$ and the other has measure $-\infty$. – hardmath Feb 2 '11 at 4:04

For signed measures assuming both $\infty$ and $-\infty$ wouldn't a right thing, for $\infty-\infty$ is undefined. With $\mathbb{C}$ the problem is that, I think, there is no ordering on it.

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