We work on the space $\mathcal{K}$ of compact subsets of $[0,1]$, with the Hausdorff metric.
Let $h:[0,1]\to \mathbb R$ be a continuous strictly increasing function with $h(0)=0$. We say that a compact subset $E$ has Hausdorff $h$-measure zero if, given $\epsilon > 0$, we can find a finite collection of intervals $I_j$ of length $l_j$ ($1\le j \le n$) such that
$$ E \subset \bigcup_{j=1}^n I_j \quad \text{but}\quad \sum_{j=1}^n h(l_j) < \epsilon. $$
The question is: show that the set $\mathcal{C}$ of compact subsets with Hausdorff $h$-measure zero is the complement of a set of first category in $\mathcal{K}$.
My attempts: This makes me recall this theorem that we can decompose $\mathbb R$ into a measure-zero set and a set of first category, by enumerating $\mathbb R$ as $\{q_1,q_2,\dots\}$ defining
$$ I_{ij} = \left(q_i-1/2^{i+j},q_i+1/2^{i+j}\right),\quad G_j = \bigcup_{i=1}^\infty I_{ij}, \quad B = \bigcap_{j=1}^\infty G_j. $$
It then follows that $B$ has measure zero and $\mathbb R\setminus B$ is of first category.
I wonder if I can do similar things to the compact subsetsin $[0,1]$ by taking $\delta_i$ such that $\sum h(2\delta_i)\to 0$ and define things like
$$ I_{ij} = \left(q_i-\delta_{i+j},q_i+\delta_{i+j} \right),\quad G_j = \bigcup_{i=1}^\infty I_{ij}, \quad B = \bigcap_{j=1}^\infty G_j， $$ and got stuck here. Any help is appreciated.