For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by $$ {\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}. $$ Then define $$ \dim_*(\mu)={\rm ess} \inf {\rm dim}_*(\mu, x), \dim^*(\mu)={\rm ess} \sup {\rm dim}_*(\mu, x), $$ and $$ {\rm Dim}_*(\mu)={\rm ess} \inf {\rm dim}^*(\mu, x), {\rm Dim}^*(\mu)={\rm ess} \sup {\rm dim}^*(\mu, x). $$

It is well-known in fractal geometry that $$ \dim_*(\mu)=\inf (\dim_H(E): \mu(E)>0), \dim^*(\mu)=\inf (\dim_H(E): \mu(E)=1), $$ and $$ {\rm Dim}_*(\mu)=\inf (\dim_P(E): \mu(E)>0), {\rm Dim}^*(\mu)=\inf (\dim_P(E): \mu(E)=1), $$ where $\dim_H$ is Hausdorff dimension and $\dim_P$ is Packing dimension.

I am trying to prove this, but I didn't find the detailed proof. If one could provide a detailed proof or a reference, I would be appreciated.

Thanks.