Analysis – HW 3 – Strong measure zero

This set is due Wednesday, October 30, at the beginning of lecture.

[Edit, Oct. 30: The original version of the problem set had some mistakes, and has been replaced accordingly.]

Recall that a set A\subseteq \mathbb R is measure zero iff for all \epsilon>0 there is a sequence (I_n\mid n\in\mathbb N) of open intervals such that \displaystyle \sum_{n\in\mathbb N}\mathrm{lh}(I_n)<\epsilon and \bigcup_n I_n\supseteq A.

Similarly, X\subseteq\mathbb R is strong measure zero iff for any sequence (\epsilon_n\mid n\in\mathbb N) of positive reals, there is a sequence (I_n\mid n\in\mathbb N) of open intervals such that \mathrm{lh}(I_n)\le\epsilon_n for all n, and \bigcup_n I_n\supseteq X. The notion is due to Borel, in 1919.

In lecture we showed that the continuous image of a measure zero set does not need to be a set of measure zero, and that the sum of two measure zero sets does not need to be a measure zero set.

As mentioned in lecture, Borel conjectured that the strong measure zero sets are precisely the countable sets. This statement turned out to be independent of the usual axioms of set theory: If the continuum hypothesis is true, the conjecture is false. On the other hand, Laver showed in 1976 that the conjecture is true in some models of set theory.

TeX source.

Pdf file.

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