The way it's usually done is as follows: $$\dim_H \mu = \inf \{ \dim_H Z \mid \mu(Z) = 1 \}.$$ You can also study box dimension of measures, but there you take an infimum over all sets $Z$ with $\mu(Z) \geq 1-\eps$1-\epsilon$, and then a limit as$\eps \epsilon \to 0$. In addition to the books Gerald mentions, you can find a comprehensive discussion of this in Dimension Theory in Dynamical Systems by Yakov Pesin, and a more introductory discussion in Chapter 4 of Lectures on Fractal Geometry and Dynamical Systems by Yakov Pesin and Vaughn Climenhaga. 1 The way it's usually done is as follows: $$\dim_H \mu = \inf \{ \dim_H Z \mid \mu(Z) = 1 \}.$$ You can also study box dimension of measures, but there you take an infimum over all sets$Z$with$\mu(Z) \geq 1-\eps$, and then a limit as$\eps \to 0\$.