... subsets of MLg,nsatisfying (6.3), the measure νJbelongs to the Lebesgue measure class. Then the ergodicity of the 122 MARYAM MIRZAKHANIaction of the mapping class group on MLg,n(Theorem ... curves in the Modg,n-orbit of γ on X ,is determined by γ. In other words, Oγ is the set of [φ(γ)] where φ: Sg,n→ X is a marking of X.Integration over the moduli space of hyperbolic surfaces. ... x)=Volwp(Mg,n(Γ, x)).By Theorem 4.1, integrating fγ, even for a compact Riemann surface,reduces to the calculation of volumes of moduli spaces of bordered Riemannsurfaces.Idea of the proof of Theorem 4.1....