  We solve the Cauchy problem for the full non-linear homogeneous Boltzmann
system of equations describing multi-component monatomic gas mixtures for
binary interactions in three dimensions. More precisely, we show existence and
uniqueness of the vector value solution by means of an existence theorem for
ODE systems in Banach spaces under the transition probability rates assumption
corresponding to hard potentials rates in the interval $(0,1]$, with an angular
section modeled by an integrable function of the angular transition rates
modeling binary scattering effects. The initial data for the vector valued
solutions needs to be a vector of non-negative measures with finite total
number density, momentum and strictly positive energy, as well as to have a
finite $L^1_{k_*}(\mathbb{R}^3)$-integrability property corresponding to a sum
across each species of $k_*$-polynomial weighted norms depending on the
corresponding mass fraction parameter for each species as much as on the
intermolecular potential rates, referred as to the scalar polynomial moment of
order $k_*$. The existence and uniqueness rigorous results rely on a new
angular averaging lemma adjusted to vector values solution that yield a Povzner
estimate with constants that decay with the order of the corresponding
dimensionless scalar polynomial moment. In addition, such initial data yields
global generation of such scalar polynomial moments at any order as well as
their summability of moments to obtain estimates for corresponding scalar
exponentially decaying high energy tails, referred as to scalar exponential
moments associated to the system solution. Such scalar polynomial and
exponential moments propagate as well.
