A tightness criterion for $\ell^\infty$-valued local martingales with infinitely many jumps is presented. It is a generalization of Ossiander's (1987) central limit theorem in the I.I.D. case. We apply it to some nonparametric inference problems for Levy measures. We propose a kind of Nelson-Aalen's estimator, and derive its asymptotic normality and efficiency. We also give a goodness-of-fit test which is asymptotically distribution free.