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We characterize probability measure with finite moment of any order in termsof the symmetric difference operators of their Fourier transforms. By using ournew characterization, we prove the continuity $f(t,v)\in C((0,\infty),L^1_{2k-2 +\alpha})$, where $f(t, v)$ stands for the density of uniquemeasure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for thehomogeneous non-cutoff Boltzmann equation, with Maxwellian molecules,corresponding to a probability measure initial datum $F_0$ satisfying \[ \int|v|^{2k-2+\alpha} dF_0(v) < \infty, 0\leq \alpha < 2,k= 2, 3, 4,\cdots \]provided that $F_0$ is not a single Dirac mass.

A characterization of probability measure with finite moment and an application to the Boltzmann equation