General relativistic Boltzmann equation, I: Covariant treatment

This series of two articles aims at dissipating the rather dense haze existing in the present literature around the General Relativistic Boltzmann equation. In this first article, the general relativistic one-particle distribution function in phase space is defined as an average of delta functions. Thereupon, the general relativistic Boltzmann equation, to be obeyed by this function, is derived. The use of either contravariant or covariant momenta leads to different, but equivalent, forms of the equation.
The results of the present article are covariant, but not manifestly covariant. The transition to a manifestly covariant treatment, on the basis of off-shell momenta, is given in the second article.