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I'm looking for an analogy to grasp the intuitive notion of size that Baire categories on $\mathbb{R}$ provides.

For instance, the cardinality of a subset of $\mathbb{R}$ provides a notion of size in terms of number of element. The Lebesgue measure of a subset of $\mathbb{R}$ provides a notion of size in terms of length.

I can get that meager sets are small in some technical sense and comeager sets are large. Even though analogies are imperfect, I'm working hard to put a word on this notion of size without success. Any suggestions ?

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  • $\begingroup$ All the three examples are measures in the "technical" sense of the word (with cardinality requiring us to define large and small sets more carefully than you did here). It follows that comeager are like sets of "full measure" (if we consider the unit interval as our space). The Baire measure, I suppose can be seen as measuring density. $\endgroup$ – Asaf Karagila Mar 19 '14 at 1:18
  • $\begingroup$ Are comeager sets of "full measure" with respect to the Baire measure ? $\endgroup$ – M.G Mar 19 '14 at 1:30
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    $\begingroup$ If meager is like being of measure zero, then comeager is like being everything but a measure zero set. What are sets whose complement has measure zero? $\endgroup$ – Asaf Karagila Mar 19 '14 at 1:33
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For me, the best source of intuition about category comes from games.

Given a set $A\subseteq \mathbb{R}$, consider the following game:

  • There are two players, $I$ and $II$.

  • They alternate playing decreasing open sets: player $I$ plays $U_1$, player $II$ plays $V_1\subsetneq U_1$, player I plays $U_2\subsetneq V_1$, . . .

  • After infinitely many moves, player $II$ wins if there is an element of $A$ inside the intersection of all the open sets played.

Then:

$A$ is comeager iff player $II$ has a winning strategy in this game; $A$ is locally meager (that is, $A\cap N$ is meager for some nonempty open $N$) iff player $I$ has a winning strategy in this game.

Basically, players $I$ and $II$ are building the decimal expansion of some real number; and player $II$ is trying to make the number they build wind up in $A$. (This isn't quite right, but oh well.) $A$ is comeager iff player $II$ can do this.


EDIT: There are also game interpretations of Lebesgue measure; they are somewhat more complicated, though. See e.g. http://matwbn.icm.edu.pl/ksiazki/fm/fm54/fm5417.pdf. (I recall a much more understandable article on the subject, but I'm having trouble tracking it down at the moment - I'll add it if I find it.)


It's also worth pointing out that there are comeager sets with Lebesgue measure zero - just an interesting point of contrast.

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  • $\begingroup$ This is an interesting perspective, thanks ! I'm glad I've resuscitated this post. $\endgroup$ – M.G Jun 2 '16 at 17:17
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I came across this quote from R. Baire (1899) today in Functional Analysis from Stein & Shakarshi :

We see the profound difference that lies between sets of the two categories; this difference lies not within denumerability, nor within density, since a set of the first category can have the power of the continuum and can also be dense in any interval one considers; but is in some sense a combination of these two preceding notions.

I felt the need to share it.

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