Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

Let $N_g$ be a closed non-orientable surface of genus $g\ge 3$. Let ${\mathrm{Homeo}}_0(N_g,\mu )$ be the identity component of the group of measure-preserving homeomorphisms of $N_g$. In this work we prove that the third bounded cohomology of ${\mathrm{Homeo}}_0(N_g,\mu )$ is infinite dimensional.