  We discuss equivariance for linear liftings of measurable functions.
Existence is established when a transformation group acts amenably, as e.g. the
Moebius group of the projective line.
  Since the general proof is very simple but not explicit, we also provide a
much more explicit lifting for semi-simple Lie groups acting on their
Furstenberg boundary, using unrestricted Fatou convergence. This setting is
relevant to cocycles for characteristic classes.
