I am asked to find a self-adjoint operator $A$ on a Hilbert space $H$ such that $[0, 1] \subseteq \sigma_{sing} (A)$.
Here, $\sigma_{sing}(A) = \sigma (A|_{H_{sing}})$.
(EDIT: The definition(s) are given here: $H_{sing} = \{ v \in H : \mu_v \mbox{ is continuous singular w.r.t. Lebesgue measure on } \mathbb{R} \}$, where $\mu_v$ is the measure induced from the positive functional $T_v : C(\sigma(A)) \to \mathbb{R}$ defined by $T_v f = \langle f(A) v, v \rangle$)
I was given a hint that find a multiplicity free (i.e. has a cyclic vector) with spectral measure as a weighted sum of translated Cantor measure.
So I guess the measure needed should be something like $$\nu = \sum_{n = 1}^\infty \frac{1}{2^n} \mu_n$$ where $\mu_n (E) = \mu (E - n)$ and $\mu$ is the self-similar probability measure on the Cantor set.
But I have no idea how to construct such $A$.
Any hints are appreciated.