Let $X\subseteq\mathbb{R}^n$ be a Borel set. Suppose $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension and $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.
Furthermore, suppose $\mathcal{H}^{\dim_{\text{H}}(X)}(X)=+\infty$ and the generalization of the Hausdorff measure (i.e., $\mathscr{H}^{\phi_{h,g}^{\mu}(q,t)}$) is defined at the end of this paper. (The generalization $\mathscr{H}^{\phi_{h,g}^{\mu}(q,t)}$ has intuitive and applicable properties. For simpler generalizations, see Bilel Selmi's papers.)
Question: For all Borel $X\subseteq\mathbb{R}^{n}$ (i.e., $\mathcal{H}^{\dim_{\text{H}}(X)}(X)=+\infty$) is there a sequence of sets $(X_j)_{j\in\mathbb{N}}$ with a set theoretic limit of $X$, such that for all $j\in\mathbb{N}$, $0<\mathscr{H}^{\phi_{h,g}^{\mu}(q,t)}(X_j)<+\infty$. (For instance, suppose $X$ is the Liouville numbers.)
Clarification: I want a generalized Hausdorff measure, since when $X$ is the Louiville numbers and we wish to find a sequence of sets $(X_j)_{j\in\mathbb{N}}$ with a set theoretic limit of $X$, such that for all $j\in\mathbb{N}$, $0<\mathcal{H}^{\text{dim}_{\text{H}}(X)}(X_j)<+\infty$, then this is impossible, since $\mathcal{H}^{0}(X_j)$ is finite only if $X_j$ is finite. (The former might work by replacing $\dim_{\text{H}}(X)$ with $\dim_{\text{H}}(X_j)$, but I'm not sure whether this would work for all Borel $X\subseteq \mathbb{R}^{n}$.)
