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When I was taught undergraduate set theory I was told that the idea for club sets is that they are "of full measure" and the non-stationary sets are "of null measure". (There were more than two people saying that it seems, giving me an impression that there might be a particular reason behind it, instead of just saying non-stationary sets are "negligible".) However I was told that in general it is not the case that P(κ)/NS is a measurable algebra, where κ is an uncountable cardinal and NS is the non-stationary ideal. On the other hand, I do see that the normality of an ideal on P(κ) gets mentioned in measure theory (see Chapter 54: Real-valued cardinals of Fremlin - Measure theory, the end of the first page), and NS is contained in every normal ideal on P(κ). So what role do normal ideals or non-stationary ideals play in measure theory?

LSpice
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sobach'e_pole
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    There is no way to assign positive numbers to all stationary sets and get a measure, because of Solovay’s stationary partition theorem. – Monroe Eskew Mar 09 '24 at 09:42

1 Answers1

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If κ has uncountable cofinality, then the collection of subsets of κ that contain or omit a club is a σ algebra, and the function assigning measure 1 or 0, respectively, is indeed a countably additive measure. The stationary sets are exactly the sets with positive outer measure. The nonstationary sets are exactly the null sets. Club sets have full measure 1. It's not just a metaphor, but rather is literally a measure and indeed a probability space.

Joel David Hamkins
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    "It's not just a metaphor, it's a measure!" sounds like an advertising slogan for some as-yet undreamt of product. – LSpice Mar 08 '24 at 19:33
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    @LSpice Sorry for that. But I have heard versions of this question from students many times, who thought that all the talk about "measure 1" and "measure 0" regarding the club filter and nonstationary sets was just an analogue, a guiding intuition using ideas about measure, rather than the real thing. – Joel David Hamkins Mar 08 '24 at 19:47
  • Re, certainly no need for apology! I was just making a joke. – LSpice Mar 08 '24 at 22:53
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    Joel, in this scenario the measure algebra is just the club filter and the non-stationary sets. P(κ)/NS is not a measure algebra. – Asaf Karagila Mar 08 '24 at 23:40
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    It seems you pointed this out before on MO many years ago: https://mathoverflow.net/a/37503/25028 – Sam Hopkins Mar 09 '24 at 03:21
  • @AsafKaragila Yes, agreed. I was mainly addressing the idea expressed in the question that talking about measure in regard to clubs and stationary sets is merely an intuitive idea, rather than literally true. The club filter provides an honest probability measure, for which the nonstationary sets are exactly the null sets. (And thanks, Sam, for pointing out that I have made this point in the past.) – Joel David Hamkins Mar 09 '24 at 09:25