Need clarification on definition of cantor distribution

Question

In https://en.wikipedia.org/wiki/Cantor_distribution it says it's a probability distribution with CDF the cantor function $C(x)$. I'm a bit confused on what the probability distribution actually is. Is the sample space here defined to be $[0,1]$, and the probability $P$ on $[0,1]$ is defined like the Lebesgue measure on $[0,1]$ but with $P([a,b])=C(b)-C(a)$, instead of $P([a,b])=b-a$ like with the usual Lebesgue measure? So $P$ is $1$ on the cantor set, and is $0$ on it's complement?