SL(2,ℝ)-invariant probability measures on the moduli spaces of translation surfaces are regular

In the moduli space H_g of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ≤ ρ. We prove that this subset of H_g has measure o(ρ²) w.r.t. any probability measure on H_g which is invariant under the natural action of SL(2,ℝ). This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunov exponents of the KZ-cocycle.