  We prove the existence and establish the Lifschitz singularity of the
integrated density of states for certain random Hamiltonians
$H^\omega=H_0+V^\omega$ on fractal spaces of infinite diameter. The kinetic
term $H_0$ is given by $\phi(-\mathcal L),$ where $\mathcal L$ is the Laplacian
on the fractal and $\phi$ is a completely monotone function satisfying some
mild regularity conditions. The random potential $V^\omega$ is of alloy-type.
