I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets.
The general setup to tease out these interpretations is as follows. Imagine you flip a coin infinitely many times, thereby picking a point $x \in 2^\omega$. If it turns out $x \in B \subseteq 2^\omega$, something Bad happens. If $B$ is "small", that should correspond to some informal notion of safety.
For example:
- If $B$ is codense, we can interpret that to mean that, as we are in the process of flipping the coin, there is always some possible hope of avoiding the bad outcome (i.e., we can't guarantee the bad outcome with only finitely many flips).
- If $B$ is nowhere dense, that means that, as we flip coins, there will always be hope that, with only finitely many more flips, we can guarantee a non-bad outcome. (In contrast, if $B$ were only codense, we might not be able to have our hopes realized until after we've seen all $\omega$ flips.)
- If $B$ is measure zero, we will almost surely avoid the bad outcome (in the sense of probability).
So, I inquire whether there is a similar informal judgment we can make if $B$ is meager.
After some more thought, I have a description. It's a bit unsatisfying, since it seems like it's just reading off the definition, but it lines up well with how we use meager sets in practice. I don't think there is a much better way of looking at it.
If $B$ is meager, that means there is some (countable) list of things that could go wrong, and the bad outcome $B$ occurs if anything on that list occurs. At any point in the coin-flipping process, there is hope that we could avoid the bad outcome in the end by eliminating each possibility on the list one-by-one. While we can hope to eliminate any individual list item with only finitely many more flips, the list may be infinite, so we may not be able to eliminate them all until "the end."