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There's a very good notion of "local" dimension of a measure at a point $x$: $$\dim_x(\mu) = \lim_{r\rightarrow 0}\frac{\mu(B_r(x))}{r}$$0}\frac{\log\mu(B_r(x))}{\log r}$$where B_r(x) is the ball of radius r centered at x. (Intuitively, we expect that in a d-dimensional space, the volume of a ball is proportional to the dth power of the radius, which immediately leads to this definition.) In general, the local dimension isn't defined everywhere and depends on x when it is, but under certain conditions, it is constant \mu-almost everywhere, in which case it makes sense to call it the dimension of the measure. In many cases, the measure \mu is more interesting than its support, and the dimension defined thusly will reflect this. For example, consider a stochastic map on the interval [0,1] that maps it affinely onto [0,1/4], [1/4,3/4], or [3/4,1], each with probability 1/3. If \mu is the invariant measure, then the support of \mu is the whole interval, but you can check that \dim_x(\mu)=\frac{3\log 3}{5\log 2} for \mu-a. e. x (if I didn't screw it up). You can also verify that there are plenty of exceptional x for which \dim_x(\mu) is something else or undefined. 1 There's a very good notion of "local" dimension of a measure at a point x:$$ \dim_x(\mu) = \lim_{r\rightarrow 0}\frac{\mu(B_r(x))}{r}
where $B_r(x)$ is the ball of radius $r$ centered at $x$. (Intuitively, we expect that in a $d$-dimensional space, the volume of a ball is proportional to the $d$th power of the radius, which immediately leads to this definition.) In general, the local dimension isn't defined everywhere and depends on $x$ when it is, but under certain conditions, it is constant $\mu$-almost everywhere, in which case it makes sense to call it the dimension of the measure.
In many cases, the measure $\mu$ is more interesting than its support, and the dimension defined thusly will reflect this. For example, consider a stochastic map on the interval $[0,1]$ that maps it affinely onto $[0,1/4]$, $[1/4,3/4]$, or $[3/4,1]$, each with probability 1/3. If $\mu$ is the invariant measure, then the support of $\mu$ is the whole interval, but you can check that $\dim_x(\mu)=\frac{3\log 3}{5\log 2}$ for $\mu$-a. e. $x$ (if I didn't screw it up). You can also verify that there are plenty of exceptional $x$ for which $\dim_x(\mu)$ is something else or undefined.