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Background :

Let us consider the Lebesgue measure space $(\Bbb{R}, \mathcal{L}(\Bbb{R}),m) $.

Here measurable set means Lebesgue measurable and measure means Lebesgue measure.

$\mathcal{S}\subset \mathcal{P}(\Bbb{R}) $ is called a class of small sets ( or a $\sigma$-ideal) if

  1. $\emptyset \in \mathcal{S}$

  2. $A\in\mathcal{S}$ and $B\subset A$ implies $B\in \mathcal{S}$

  3. $\bigcup_{n\in \Bbb{N}} A_n\in\mathcal{S}$ whenever $A_n\in\mathcal{S}$ for all $n\in \Bbb{N}$

The class of countable sets, null sets, meager sets are all small in one sense and other.

In this thread, we want to study the relationship between sets which are small in sense of cardinality and measure.

  1. Small in sense of cardinality implies small in sense of measure:

$A\subset \Bbb{R}$ is countable implies $m(A)=0$.

  1. Small in sense measure may not implies small in sense of cardinality:

Example: The Cantor set.

Question: Is there any condition that makes a measure zero set necessarily countable?

UPDATE:

Strong measure zero set: $A\subset \Bbb{R}$ is strong measure $0$ set if for every sequence $(\delta_n)\subset \Bbb{R}^+$ there exists a sequence $(I_n)$ of intervals such that $\ell(I_n) < \delta_n$ for all $n$ and $A$ is contained in the union of the $I_n$.

From the definition, it is clear that a strong measure zero set is a null set.

The Cantor set is a measure $0$ set but fails to be countable.But the Cantor set is not strong measure zero set(see here).

In a celebrated paper,Borel conjectured that every Strong measure zero set of reals was countable. This statement known as Borel Conjecture.

It is now known that this statement is independent of ZFC.

Sourav Ghosh
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    For a meager set with positive measure look at a fat Cantor set. For relationships between measure and category the book by Oxtoby appropriately titled "measure and category" has a lot of useful informations. As for conditions that make a measure zero set countable I cannot think of any nontrivial one right now, but you might be interested in the Borel conjecture – Alessandro Codenotti Dec 06 '21 at 15:22
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    https://en.wikipedia.org/wiki/Strong_measure_zero_set – Noah Schweber Dec 06 '21 at 15:25
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    One could simply define a measure where points have non-zero measures. – herb steinberg Dec 06 '21 at 17:13
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    @Alessandro But that is just an uncountable Polish space, so of second category. The way the question is written it seems that we are looking for a set that is meager in its subspace topology. – Jonathan Schilhan Dec 06 '21 at 17:23
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    It is spelled "Category" as in categorising things (Baire use the terms 1st and 2nd category as in a shelve, sort of things). These terms provide no insight in the meaning, so people called them meager/nowhere-dense and non-meager/somewhere-dense. – William M. Dec 06 '21 at 17:33
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    Also, there are versions of the Cantor set that have positive measure (called "fat Cantor set") while retaining their topological properties (e.g. still meager). The three properties (cardinality, measure and meagerness) really measure different things. – William M. Dec 06 '21 at 17:36
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    This answer might be of interest. Also, the property that the closure of the set has measure zero (and not just the set itself has measure zero) is a significant strengthening of being small. Indeed, any such set is also first category (nowhere dense, in fact), and there exist sets simultaneously measure zero and first category that cannot be the subset of any countable union of closed measure zero sets (e.g. see this answer). – Dave L. Renfro Dec 06 '21 at 18:50
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    "is there any condition that makes a measure zero set necessarily countable?" Borel 1919 conjectured such a thing, now known as strong measure zero. See https://en.wikipedia.org/wiki/Strong_measure_zero_set for today's information. – GEdgar Jan 21 '22 at 17:29
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    I deleted comment and retracted sorry I didn't check how old it was – SBK Aug 15 '22 at 18:54

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