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