  We investigate the transformation of the distribution function in the
relativistic case, a problem of interest in plasma when particles with high
(relativistic) velocities come into play as for instance in radiation belt
physics, in the electron-cyclotron maser radiation theory, in the vicinity of
high-Mach number shocks where particles are accelerated to high speeds, and
generally in solar and astrophysical plasmas. We show that the phase-space
volume element is a Lorentz constant and construct the general particle
distribution function from first principles. Application to thermal equilibrium
lets us derive a modified version of the isotropic relativistic thermal
distribution, the modified J\"uttner distribution corrected for the
Lorentz-invariant phase-space volume element. Finally, we discuss the
relativistic modification of a number of plasma parameters.
  Keywords: Relativistic invariance of the one-particle distribution function,
J\"uttner distribution, relativistic plasma parameters
