Local dimension of measures

Question

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.

Answer 1

Some of the relevant considerations can be found in the book [1] and the classic paper [2]. Look e.g. at Lemma 1.4.4 in [2] for $\dim^*$. Billingsley's lemma, that you can find on [1] or in [3], Lemma 1.4.1, page 16, easily yields both statements about $dim^*$ and $dim_*$. For the statements involving packing dimension, see the original papers [4], [5] or [6], [7] for another approach.

[1] Billingsley, Patrick. Ergodic theory and information. Wiley, NY 1965.

[2] L.S. Young, Dimension, entropy and lyapunov exponents. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/dimension-entropy-and-lyapunov-exponents/5B6962A34BACD4A07EA5C7B6AE539051

[3] Bishop, C., & Peres, Y. (2016). Fractals in Probability and Analysis (Cambridge Studies in Advanced Mathematics). Cambridge University Press. PDF at https://www.math.stonybrook.edu/~bishop/classes/math324.F15/book1Dec15.pdf

[4] Tricot, C., 1982, Two definitions of fractional dimension. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 91, No. 1, pp. 57-74). Cambridge University Press.

[5] Taylor, S. James, and Claude Tricot. "Packing measure, and its evaluation for a Brownian path." Transactions of the American Mathematical Society 288, no. 2 (1985): 679-699.

[6] Cutler, Colleen D. "Strong and weak duality principles for fractal dimension in Euclidean space." In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 118, no. 3, pp. 393-410. Cambridge University Press, 1995.

[7] Cutler, Colleen D. "A review of the theory and estimation of fractal dimension." Dimension estimation and models (1993): 1-107.