Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets $$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{doesn't}\;\mathrm{exist}\right\}$$ where $d_\mu(x)$ denotes local dimension (or consider local entropy or Birkhoff Averages). We can easily say something about the dimension spectrum, namely $$\dim_H(\left\{x\mid d_\mu(x)=\alpha\right\}).$$

Can anything be said about $$\bigcup_{\alpha>0}\left\{x\mid d_\mu(x)=\alpha\right\}?$$ Moreover can we obtain the Hausdorff dimension of this (uncountable) union.