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 $\eps \to 0$.

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.

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 $\mu(Z) \geq 1-\epsilon$, and then a limit as $\epsilon \to 0$.

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.