# Local dimension of measures

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.

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

 Billingsley, Patrick. Ergodic theory and information. Wiley, NY 1965.

 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

 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

 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.

 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.

 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.

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