Intuition for meagre sets

Question

So I'm putting together a presentation in topology on BCT and I have the following theorems:

Let $D$ be the set of somewhere differentiable functions $[0,1] \to \mathbb R$. Then $D$ is meagre in $C[0,1]$.

Let $f \colon [0,1] \to \mathbb R$ be a differentiable function with derivative $f'$. Let $X$ be the set of points for which $f'$ is discontinuous. Then $X$ is meagre in $[0,1]$.

My temptation to rationalise these intuitively (as a taster in the introduction) as follows:

Most continuous functions are nowhere differentiable.

Derivatives are continuous at most points.

This is based off my understanding that a set $A$ is meagre in $B$ if it is small relative to $B$ in a precise sense. But there are instances where a meagre set can be quite large (even of full measure), and I've been told that it's not really correct to say "most" in this context. However, I've seen quite a lot of posts on here giving similar intuitive restatements.

I thought I'd ask here for clarification - is it proper/well-understood to say "most" here? I could say "many" but that doesn't really feel as punchy. If it's not proper to say most here, is there any better way to put it? My topological intuition is still fairly weak.

Answer 1

There are multiple notions of "mostness" out there. Measure and category each provide their own notions of "most" - namely, "full measure" and "comeager," respectively - and these generally do not agree: there are null sets which are comeager and there are meager sets which have full measure.

Whether one should use the term "most" in a category-based sense depends on context. Certainly terms like "almost every" are exclusively connected to measure, while "generic" is exclusively connected to category, but "most" is somewhat flexible. I would personally say that in analysis, "most" should be reserved for the measure-theoretic sense, primarily because category plays a significantly lesser role there; however, in general topology I would take the opposite stance, since category makes sense in arbitrary topological spaces while measure doesn't.

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Re: what terminology you should use, let me go back to the term "generic" I mentioned above. This is a rather subtle notion; intuitively, a generic point is a point which doesn't lie in any "simple" meager set. Of course this isn't an absolute notion, but it's often used as a weasel-word when a comeager set is "hidden in the background." So for example one might say

A generic function is nowhere differentiable

or

Given a differentiable function $f$, the derivative $f'$ is continuous at a generic point.

Unfortunately, this terminology is somewhat uncommon. It's also frequently grammatically unwieldy: in the second example above it would sound better to say e.g. "The derivative of a differentiable function is generically continuous," but I've only rarely seen this usage outside of logic.

Ultimately since measure gets talked about far more than category in most areas of mathematics (in my experience anyways), it gets the bulk of the "simple language" descriptions.