Small sets on $\mathbb{R}$. I was thinking of different definitions of small subsets on $\mathbb{R}$, such as meagre or zero-measure. These are quite well-known, so I was searching for different notions. 
Define a set has zero-content if for every $\epsilon > 0$, the set can be covered by a finite number of intervals with total length less than $\epsilon$.
A set is unimportant if it is a countable union of zero-content sets.
Slightly different: a subset of $\mathbb{R}$ is called tiny if for every $\epsilon > 0$ there exist a sequence of intervals $I_1, I_2, \ldots$ that cover the set such that for every $i$, $|I_i| \leq \epsilon^i$.
I have a few questions concerning these definitions:


*

*Does there exist an uncountable unimportant set? 

*I already found every unimportant set is of zero-measure, but is the converse true? I guess not. 

*What is the relation between tiny and zero-measure?


Thanks in advance!
 A: On tiny sets: It was already pointed out that tiny sets are null and the classical Cantor set is not tiny. Also, there are uncountable tiny sets. To see this, construct a Cantor scheme of closed intervals in $[0, 1]$ such that the $n$th level of your tree is a disjoint union of $2^n$ intervals each of length less than $\displaystyle (1/n)^{2^{n}}$. The resulting perfect set $C$ is a size continuum tiny set. A similar notion that leads to deeper problems is that of a strongly null set.
On (2): A null dense $G_{\delta}$ set $X \subseteq \mathbb{R}$ cannot be covered by countably many sets of zero (Jordan) content.
A: Tiny sets have zero Hausdorff dimension. Thus any set with positive Hausdorff dimension (certain cantor sets, etc.) cannot be tiny.
The Hausdorff dimension of a set $A \subset \mathbb R$ may be defined via the following process:


*

*For $s \ge 0$ and $\epsilon > 0$ define $\displaystyle H^s_\epsilon(A) = \inf \left\{ \sum_k |I_k|^s : A \subset \cup I_k,\ |I_k| < \epsilon \right\}$.

*Since $H^s_\epsilon(A)$ is increasing as $\epsilon \to 0^+$, the quantity $H^s(A) \displaystyle = \sup_{\epsilon > 0 } H^s_\epsilon(A) = \lim_{\epsilon \to 0^+} H^s_\epsilon(A)$ exists and is termed the Hausdorff $s$-measure of $A$. It turns out that $H^s$ is in fact a Borel measure on $\mathbb R$.

*There is a unique value $s_0$ with the property that $H^s(A) = \infty$ for all $0 \le s < s_0$ and $H^s(A) = 0$ for all $s > s_0$. This number is the Hausdorff dimension of $A$.


If $A$ is tiny, then for any $s > 0$ and $\epsilon > 0$ there exist intervals $I_k$ with the property that $A \subset \cup I_k$ and $|I_k| < \epsilon ^k$. Thus
$$ H^s_\epsilon(A) \le \sum_k |I_k|^s \le \sum_k \epsilon^{sk} = \frac{\epsilon^s}{1 - \epsilon^s}.$$
Let $\epsilon \to 0^+$ to obtain $H^s(A) = 0$. This is the case for any positive $s$, forcing the dimension of $A$ to be $0$.
