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The real analysis book that I had read mentions that "most continuous functions do not have derivatives at any point." The precise mathematical definition of "most" can be inferred from the following theorem.

Theorem: Let $C[0,1]$ be the collection of continuous functions on the closed interval $[0, 1]$ and let the set $D$ be defined as $$D = \{f \in C[0,1] : f'(x) \text{ exists for some} x ∈ [0,1]\}.$$ Then $D$ is a set of first category (a meager set) in $C[0,1]$.

From the above theorem, the qualification of "most" (referring to the sentence mentioned in the very first line of this post) seems to imply that it means "non-meagerness" of the set of continuous nowhere-differentiable functions. So a meager set should then be equivalent to a negligible set. As regards to the source where the term negligible is used to qualify a meager set, Wikipedia is an example among numerous others. I would like to know how a meager set can qualify as being negligible, given the following facts.

  1. There are meager sets with non-zero Lebesgue measure like the fat-Cantor set.
  2. There are meager sets whose cardinality is same as that of the real numbers, like in the Cantor set.

So meager sets can have as many elements in them as that in the set of reals, and can have a non-zero measure, but are still considered negligible simply because they are nowhere dense?

I would like to know the justification of referring to meager sets as being negligible especially in light of the above two facts that I have mentioned about meager sets.

TryingHardToBecomeAGoodPrSlvr
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    The use of the word "negligible" here is not rigorous. It's just being used here to give you the idea that the word "meager" in some sense means "small." There are of course many ways I could say that a subset of $\mathbb R$ is "small." It could have zero Lebesgue measure, it could be countable, it could be a nowhere dense $F_\sigma$ set (which is strictly weaker than being meager). These are all reasonable definitions of small that give the right answer in some intuitive cases, but the interesting this is to see how they interact in the edge cases, as you have considered in your post. – D. Brogan Aug 27 '21 at 14:17
  • @D.Brogan Many sources tend to qualify properties in a non-rigorous manner as you have rightly pointed out. This was my hunch as well. However, pure math was the last place I'd have expected something to be not rigorous :). So I take it that they consider it to be small or negligible simply because of the nowhere dense nature of these sets? – TryingHardToBecomeAGoodPrSlvr Aug 27 '21 at 14:21
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    Pure mathematicians use natural human language to impart intuition all the time. Helping our readers to understand the mathematics by imparting intuition is just as important as providing rigorous proofs. – Lee Mosher Aug 27 '21 at 14:47
  • There are many different notions of "small" or "negligible" in mathematics, just as there are many different notions of "number", "space", "equivalent", and many other general-use terms. Regarding your comment seems to imply that it means "non-meagerness", this is true, but that's like saying "a number that's greater than $1000$ is nonzero" (i.e. a huge loss in content occurs in drawing this conclusion from the premise). (continued) – Dave L. Renfro Aug 27 '21 at 18:28
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    For the distinction between meager, not meager, and complement is meager, see my answer to Discontinuity set of derivative. For more about the nowhere differentiability behavior of "most continuous functions", see my answer to Generic Elements of a Set.. Finally, my answer to Size of a function space might also be of interest (requires reputation $\geq 3000).$ – Dave L. Renfro Aug 27 '21 at 18:28
  • @DaveL.Renfro Thanks for the links to your answers! I will read them with interest. – TryingHardToBecomeAGoodPrSlvr Aug 27 '21 at 18:31

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Oftentimes it is useful to call a set negligible if it is a member of some specific proper $\sigma$-ideal. A family $\mathcal{I} \subseteq \mathcal{P}(X)$ is called a proper $\sigma$-ideal of subsets of a fixed set $X$ provided that:

  1. $\varnothing \in \mathcal{I}, X \notin \mathcal{I}$
  2. $(\forall A \subseteq X)(\forall B \in \mathcal{I}) \big( A \subseteq B \implies A \in \mathcal{I} \big)$
  3. $\displaystyle (\forall A_1, A_2, \ldots \in \mathcal{I}) \, \bigcup_{i=1}^{\infty} A_i \in \mathcal{I}$

Practically it means that if you take any countable family of negligible sets, they can not cover the whole space. Of course if you then take this uncovered residue, it still can not be covered with countably many negligible sets, and so on. This is usually enough to justify referring to a negligible set as "almost nothing" and to the complement - "almost everything".

Three classic examples of $\sigma$-ideals are:

  1. The countable subsets of an uncountable set
  2. The first Baire category subsets of $\mathbb{R}$ (i.e. meager sets)
  3. The Lebesgue measure zero subsets of $\mathbb{R}$ (i.e. null sets)

Somewhat paradoxical is the fact you stated: a set negligible in one sense might be very big in another. A profound example is the following theorem:

Theorem. The real line $\mathbb{R}$ can be expressed as a union of a meager set and a null set.

That's why formally it is better to speak of e.g. $\mathcal{M}$-negligible sets (meager sets) so that a countable union of $\mathcal{M}$-negligible sets is again $\mathcal{M}$-negligible. But usually it is understood which kind of negligibility is meant so that disambiguation is not necessary.

Anyway: the notion of a $\sigma$-ideal seems to capture everything needed to consider a set "small" or "negligible", and that is why meager sets are called so.

Adayah
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