What does this mean "A stationary set is analogous to a set of non-zero measure in measure theory"? Can we have a similar comparison for club sets too?

(Plus, I don't know how much it is meaningful if I ask is there any idea behind this analogy, I mean why the author has compared a subset of a cardinal with a measurable subset in measure theory?)


Given a complete measure space $X$, consider the collection $F$ of full measure sets. These are the sets whose complement has measure 0.

Note that if $Y\in F$ then any superset of $Y$ (that is, any $Z$ with $Y\subseteq Z\subseteq X$) is in $F$ as well. (This is why I require the measure to be complete, which means that any subset of a measure zero set is measurable---and therefore of measure 0.)

Also, $X\in F,\emptyset\notin F$, and if $A$ and $B$ are in $F$ then so is their intersection. In fact, $F$ is closed under countable intersections.

This means that $F$ is a $\sigma$-complete filter. The members of $F$ are "big" (in the sense of the measure). Their complements are small (one even refers sometimes to measure zero sets as null or negligible). The sets of positive measure are thus those that are not small. They do not necessarily belong to $F$, but they are not null either.

It is in this sense that the notions are analogous: Club sets give us a notion of largeness (the analogue of full measure sets). Indeed, given $\kappa$ regular, the subsets of $\kappa$ that contain a club form a $\sigma$-complete filter (actually, a $\kappa$-complete filter). The stationary sets are those that are not disjoint from any club. Thus, they are not negligible in this interpretation.

Any reasonable filter will give you a similar analogous notion of largeness: large sets are those in the filter. Negligible, or small sets are their complements. Those that are not negligible play the role of stationary sets (or of sets of positive measure).

This is a common theme in analysis, where we also study the collection of comeager sets. Analysts study several other filters of sets as well. There is no universal notion of largeness, of course, what filter to consider is definitely context dependent. In infinitary combinatorics, the club filter has shown over and over again to be central.

  • $\begingroup$ Thank you so much. Does exist any other example for the usage of the dual notion of filter? $\endgroup$ Feb 7 '20 at 18:03
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    $\begingroup$ Maximal filters (ultrafilters) play a significant role in set theory. Both at small cardinals (even $\omega$) and at large cardinals, where they refine the notions discussed above. In analysis the two more prominent examples are measure and category, but there are many others. Perhaps the best known come from the study of the "density topology" and its generalizations. See for example here. Note that in many cases people refer directly of the ideal of small sets rather than the filter of large sets. $\endgroup$ Feb 7 '20 at 18:19

The idea is that in the interval $[0,1]$ a set is "almost everything" if and only if it has measure $1$, and a measurable set has positive measure if and only if it meets every measure $1$ set.

In that sense, clubs are measure $1$ set. So being a club is being almost everything. In this sense, for example, a "typical countable ordinal" is a limit ordinal, and in fact a limit of limit ordinals, and in fact an admissible ordinal, and in fact a limit of limits of limits of admissible ordinals, and so on.

Stationary sets are exactly those who meet every club, that is, sets that meet every set of measure $1$. So those are sets which are not contained in a null set. So these are the sets of positive measure.

  • $\begingroup$ Thank you so much. This may seem irrelevant, but do set theorists have a subset of a cardinal or an ordinal which is analogous to non-measurable sets in the measure theory, like the one we have for the Lebesgue measure? ( In the presence of the axiom of choice or in the absence of that ) $\endgroup$ Feb 7 '20 at 18:16
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    $\begingroup$ No, this is more of an outer measure situation than a Lebesgue measure situation. $\endgroup$
    – Asaf Karagila
    Feb 7 '20 at 18:16
  • $\begingroup$ @AsafKaragila hi, sorry to bother, but I was actually very curious whether it should be "set theory or related topics" instead of "set theory and related topics" in your bio (or whatever it's called). $\endgroup$ Feb 22 '20 at 14:41
  • $\begingroup$ @mathworker21: Does it really matter? $\endgroup$
    – Asaf Karagila
    Feb 22 '20 at 15:19
  • $\begingroup$ @AsafKaragila If I want to email you about something just about set theory (and not related topics), then your bio says I shouldn't. $\endgroup$ Feb 22 '20 at 18:33

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