  We generalize the notions of definable amenability and extreme definable
amenability to continuous structures and show that the stable and ultracompact
groups are definable amenable. In addition, we characterize both notions in
terms of fixed-point properties. We prove that, for dependent theories,
definable amenability is equivalent to the existence of a good S1 ideal.
Finally, we show the randomizations of first-order definable amenable groups
are extremely definably amenable.
