For any positive real number $s$, define an $s$-nullset as a subset $A$ of the real line such that, for any $\epsilon > 0$, there exists a sequence of intervals $\{I_n\}_{n=1}^{\infty}$ having the following three properties:

  1. $A \subseteq \cup_{n=1}^\infty I_n\;;$
  2. $|I_n| < \epsilon, \forall \; n \in \{1,2,3,\dots\}\;;$ and
  3. $\sum_{n=1}^{\infty}|I_n|^s < \epsilon$.

It is not difficult to show that, for any $0 < s < 1$, an $s$-nullset is also a $1$-nullset. I want to know how to prove that the converse is not true, namely, that for every $0 < s < 1$ there is a $1$-nullset that is not an $s$-nullset.


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