I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2. Such spaces are also known as standard probability spaces but the definitions are not unique in the literature.

Let us restrict to probability measures.

Theorem 9.4.7: If $(M, \mathcal{M}, \mu)$ is Lebesgue-Rokhlin then it is isomorphic mod0 to $([0,1], \mathcal{B}[0,1], \nu)$ where $\nu = c \lambda + \sum_{n=1}^\infty \alpha_n \delta_{1/n}$ where $\alpha_n = \mu(a_n)$ and $a_n$ is the countable family of atoms of $\mu$.

Now every Borel probability measure on $[0,1]$ can be decomposed into an absolutely continuous part, a singular continuous part and an atomic part. In the above theorem I am somehow missing the singular continuous part. Is it not explicitely mentioned or what is the intuition that one only considers the absolutely continuous part and the atomic part? Is it "hidden" behind the Lebesgue measure?