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I first came across $\sigma$-algebras in measure theory. I understand that the issue is that we want to define the sets we can give a measure to, and that if we can give a set a measure, then its complement should have the "rest" of the measure assigned to the universe. I don't really understand where union comes into this; I can't just add the measure of two non-disjoint sets to get the measure of their union, yet the definition of $\sigma$-algebras expects that the union be measurable. What's the idea I'm missing here?

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  • $\begingroup$ The inclusion-exclusion principle tells you what the measure of the union is, provided you know the measure of the intersection. Do you believe that the intersection of measurable sets should be measurable? $\endgroup$ – Zhen Lin Aug 10 '14 at 19:09
  • $\begingroup$ I have exactly the same problem with the intersection. Inclusion-exclusion only helps to calculate the measure, it doesn't tell me why it should exist. $\endgroup$ – G. Bach Aug 10 '14 at 19:11
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    $\begingroup$ @G.Bach (If I am mistaken, please correct me.) We would like for unions and intersections to be measurable, so we define $\sigma$-algebras that way. $\endgroup$ – angryavian Aug 10 '14 at 19:21
  • $\begingroup$ @angryavian That's a perspective that I hadn't considered, thank you! $\endgroup$ – G. Bach Aug 12 '14 at 20:55

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