Let's modify Cantor set. Let $f_{\gamma}$ be a function that maps interval $[x,x+d]$ to $[x,x+d\gamma]\cup[x+d(1-\gamma),x+d\gamma]$, i.e. it removes middle part of length $d(1-2\gamma)$ (we consider $\gamma < 1/2$) and $g_{\gamma}$ puts everything but the middle part in empty interval.
Then suppose the following algorithm: at the $j$-th step we apply function $f_{\gamma}$ to all non-empty intervals, and $g_{\gamma}$ to all empty intervals. For example, after first step we have ($\gamma = 1/3$) $[0,1/3]\cup[2/3,1]$, and after second step we have $[0,1/9]\cup [2/9,4/9]\cup[5/9,7/9] \cup [8/9,1]$ and so on.
One can see that measure of set $\mathcal{C}$ is $1-\gamma$. More than that, this set is a disjoint union of Cantor set and set contained in "gaps" of Cantor set. Since gaps can be as small as we want, in any interval $[a,b]$ there is one, but this gap contains a copy of cantor set (rescaled), and if we choose $\gamma > 0$ then $\mu\left(\mathcal{C}\cap [a,b]\right) < b-a$ as $[a,b]$ contains gap, which is filled by rescaled copy of $\mathcal{C}$ and it does not have a full measure.
Am I right in my speculations? If now, how can one modify this argument to get a set of given measure, s.t. $0<\mu\left(\mathcal{C}\cap [a,b]\right) < b-a$?