Analogy for Baire categories?

Question

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 ?

Answer 1

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.

Answer 2

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.

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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.)

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