A diffeomorphism f of a compact manifold M is called almost Axiom-A if it is hyperbolic in a neighborhood of some compact f-invariant set $\Omega$, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure and such that for every point in this set the stable and unstable leaves are ‘long enough’, then f admits either a probability Sinai–Ruelle–Bowen (SRB)-measure or a $\sigma$-finite SRB-measure.