  We define $g$-expectation of a distribution as the infimum of the
$g$-expectations of all the terminal random variables sharing that
distribution. We present two special cases for nonlinear $g$ where the
$g$-expectation of distributions can be explicitly derived. As a related
problem, we introduce the notion of law-invariant $g$-expectation and provide
its sufficient conditions. Examples of application in financial dynamic
portfolio choice are supplied.
