What is the measure zero of uncountable set .

Question

Recently I was reading Methods of Real Analysis by Goldberg and had the following question.

7.1 Corollary: Every countable subset of $\mathbb{R}$ has measure zero.

How can we describe the uncountable subsets of $\mathbb{R}$ which have measure zero? E.g. the Cantor set, which is a decreasing nested sequence.

Are there any examples of uncountable subsets of $\mathbb{R}$ with measure zero which are not the limits of a decreasing nested sequence of sets? Any which are the limit of an increasing nested sequence?

Answer 1

I believe that the answer is "in some sense" no, and that this follows from the definition of a set with measure zero. See: https://en.wikipedia.org/wiki/Null_set

The intuitive idea is that any subset of the real line with measure zero can be covered by open intervals whose "total length" can be arbitrarily small.

If we take $\epsilon = \frac{1}{n}$, then any measure zero set is either the limit of a decreasing nested sequence of open interval covers (whose "total length" is $\frac{1}{n}$ for each $n$) in which case the set is Borel measurable,

or it is a proper subset of a measure-zero Borel set, so not the limit of the decreasing nested sequence, but a subset of a limit of a decreasing nested sequence.

If any set of measure zero were the limit of an increasing nested sequence of sets, all of the sets in the sequence would themselves have to be measure zero.

However, it may be possible to find such an increasing sequence of uncountable sets, all of which have measure zero, such that their limit is again an uncountable set with measure zero.

So the answer depends on which nested sequences you are willing to consider.